From 0345245e860375a32c9a437c4a9d9cae807134e9 Mon Sep 17 00:00:00 2001 From: Shashank Date: Mon, 29 May 2017 12:40:26 +0530 Subject: CMSCOPE changed --- .../special_functions/help/en_US/addchapter.sce | 11 + modules/special_functions/help/en_US/amell.xml | 65 ++++ modules/special_functions/help/en_US/bessel.xml | 346 +++++++++++++++++++++ modules/special_functions/help/en_US/beta.xml | 124 ++++++++ modules/special_functions/help/en_US/calerf.xml | 93 ++++++ modules/special_functions/help/en_US/dawson.xml | 88 ++++++ modules/special_functions/help/en_US/delip.xml | 146 +++++++++ modules/special_functions/help/en_US/dlgamma.xml | 84 +++++ modules/special_functions/help/en_US/erf.xml | 96 ++++++ modules/special_functions/help/en_US/erfc.xml | 84 +++++ modules/special_functions/help/en_US/erfcx.xml | 85 +++++ modules/special_functions/help/en_US/erfi.xml | 88 ++++++ modules/special_functions/help/en_US/erfinv.xml | 89 ++++++ modules/special_functions/help/en_US/findm.xml | 26 ++ modules/special_functions/help/en_US/gamma.xml | 105 +++++++ modules/special_functions/help/en_US/gammaln.xml | 73 +++++ modules/special_functions/help/en_US/legendre.xml | 230 ++++++++++++++ modules/special_functions/help/en_US/percentk.xml | 59 ++++ modules/special_functions/help/en_US/percentsn.xml | 100 ++++++ .../special_functions/help/fr_FR/addchapter.sce | 11 + modules/special_functions/help/fr_FR/amell.xml | 57 ++++ modules/special_functions/help/fr_FR/calerf.xml | 78 +++++ modules/special_functions/help/fr_FR/delip.xml | 134 ++++++++ modules/special_functions/help/fr_FR/dlgamma.xml | 83 +++++ modules/special_functions/help/fr_FR/erf.xml | 88 ++++++ modules/special_functions/help/fr_FR/erfc.xml | 74 +++++ modules/special_functions/help/fr_FR/erfcx.xml | 75 +++++ modules/special_functions/help/fr_FR/erfinv.xml | 80 +++++ modules/special_functions/help/fr_FR/gammaln.xml | 76 +++++ .../special_functions/help/ja_JP/addchapter.sce | 11 + modules/special_functions/help/ja_JP/amell.xml | 66 ++++ modules/special_functions/help/ja_JP/bessel.xml | 322 +++++++++++++++++++ modules/special_functions/help/ja_JP/beta.xml | 126 ++++++++ modules/special_functions/help/ja_JP/calerf.xml | 94 ++++++ modules/special_functions/help/ja_JP/dawson.xml | 87 ++++++ modules/special_functions/help/ja_JP/delip.xml | 150 +++++++++ modules/special_functions/help/ja_JP/dlgamma.xml | 86 +++++ modules/special_functions/help/ja_JP/erf.xml | 94 ++++++ modules/special_functions/help/ja_JP/erfc.xml | 83 +++++ modules/special_functions/help/ja_JP/erfcx.xml | 84 +++++ modules/special_functions/help/ja_JP/erfi.xml | 88 ++++++ modules/special_functions/help/ja_JP/erfinv.xml | 88 ++++++ modules/special_functions/help/ja_JP/findm.xml | 27 ++ modules/special_functions/help/ja_JP/gamma.xml | 108 +++++++ modules/special_functions/help/ja_JP/gammaln.xml | 77 +++++ modules/special_functions/help/ja_JP/legendre.xml | 181 +++++++++++ modules/special_functions/help/ja_JP/oldbessel.xml | 186 +++++++++++ modules/special_functions/help/ja_JP/percentk.xml | 61 ++++ modules/special_functions/help/ja_JP/percentsn.xml | 80 +++++ .../help/mml/bessel_equation1.mml | 71 +++++ .../help/mml/bessel_equation2.mml | 71 +++++ .../help/mml/bessel_equation3.mml | 98 ++++++ .../special_functions/help/mml/beta_equation1.mml | 94 ++++++ .../help/mml/dlgamma_equation1.mml | 58 ++++ .../special_functions/help/mml/gamma_equation1.mml | 50 +++ .../help/mml/legendre_equation1.mml | 77 +++++ .../help/mml/legendre_equation2.mml | 113 +++++++ .../special_functions/help/pt_BR/addchapter.sce | 11 + modules/special_functions/help/pt_BR/amell.xml | 66 ++++ modules/special_functions/help/pt_BR/bessel.xml | 256 +++++++++++++++ modules/special_functions/help/pt_BR/beta.xml | 106 +++++++ modules/special_functions/help/pt_BR/calerf.xml | 78 +++++ modules/special_functions/help/pt_BR/delip.xml | 143 +++++++++ modules/special_functions/help/pt_BR/dlgamma.xml | 71 +++++ modules/special_functions/help/pt_BR/erf.xml | 78 +++++ modules/special_functions/help/pt_BR/erfc.xml | 75 +++++ modules/special_functions/help/pt_BR/erfcx.xml | 68 ++++ modules/special_functions/help/pt_BR/erfinv.xml | 83 +++++ modules/special_functions/help/pt_BR/gamma.xml | 92 ++++++ modules/special_functions/help/pt_BR/gammaln.xml | 68 ++++ modules/special_functions/help/pt_BR/legendre.xml | 189 +++++++++++ .../special_functions/help/ru_RU/addchapter.sce | 11 + modules/special_functions/help/ru_RU/amell.xml | 65 ++++ modules/special_functions/help/ru_RU/delip.xml | 146 +++++++++ 74 files changed, 7085 insertions(+) create mode 100755 modules/special_functions/help/en_US/addchapter.sce create mode 100755 modules/special_functions/help/en_US/amell.xml create mode 100755 modules/special_functions/help/en_US/bessel.xml create mode 100755 modules/special_functions/help/en_US/beta.xml create mode 100755 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/dev/null +++ b/modules/special_functions/help/en_US/addchapter.sce @@ -0,0 +1,11 @@ +// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +// Copyright (C) 2009 - DIGITEO +// +// This file must be used under the terms of the CeCILL. +// This source file is licensed as described in the file COPYING, which +// you should have received as part of this distribution. The terms +// are also available at +// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt + +add_help_chapter("Special Functions",SCI+"/modules/special_functions/help/en_US",%T); + diff --git a/modules/special_functions/help/en_US/amell.xml b/modules/special_functions/help/en_US/amell.xml new file mode 100755 index 000000000..ad77edbdb --- /dev/null +++ b/modules/special_functions/help/en_US/amell.xml @@ -0,0 +1,65 @@ + + + + + amell + Jacobi's am function + + + Calling Sequence + [sn]=amell(u,k) + + + Arguments + + + u + + real scalar or vector + + + + k + + scalar + + + + sn + + real scalar or vector + + + + + + Description + + Computes Jacobi's elliptic function am(u,k) + where k is the parameter and u is the argument. If u + is a vector sn is the vector of the (element wise) computed values. + Used in function %sn. + + + + See Also + + + delip + + + %sn + + + + diff --git a/modules/special_functions/help/en_US/bessel.xml b/modules/special_functions/help/en_US/bessel.xml new file mode 100755 index 000000000..f90086fdc --- /dev/null +++ b/modules/special_functions/help/en_US/bessel.xml @@ -0,0 +1,346 @@ + + + + + besseli + Modified Bessel functions of the first kind (I sub + alpha). + + + + besselj + Bessel functions of the first kind (J sub alpha). + + + besselk + Modified Bessel functions of the second kind (K sub + alpha). + + + + bessely + Bessel functions of the second kind (Y sub + alpha). + + + + besselh + Bessel functions of the third kind (aka Hankel + functions) + + + + Calling Sequence + y = besseli(alpha,x [,ice]) + y = besselj(alpha,x [,ice]) + y = besselk(alpha,x [,ice]) + y = bessely(alpha,x [,ice]) + y = besselh(alpha,x) + y = besselh(alpha,K,x [,ice]) + + + + Arguments + + + x + + real or complex vector. + + + + alpha + + real vector + + + + ice + + integer flag, with default value 0 + + + + K + + integer, with possible values 1 or 2, the Hankel function + type. + + + + + + + Description + + + + besseli(alpha,x) computes modified Bessel + functions of the first kind (I sub alpha), for real order + alpha and argument x. + besseli(alpha,x,1) computes + besseli(alpha,x).*exp(-abs(real(x))). + + + + + besselj(alpha,x) computes Bessel functions of + the first kind (J sub alpha), for real order alpha + and argument x. + besselj(alpha,x,1) computes + besselj(alpha,x).*exp(-abs(imag(x))). + + + + + besselk(alpha,x) computes modified Bessel + functions of the second kind (K sub alpha), for real order + alpha and argument x. + besselk(alpha,x,1) computes + besselk(alpha,x).*exp(x). + + + + + bessely(alpha,x) computes Bessel functions of + the second kind (Y sub alpha), for real order alpha + and argument x. + bessely(alpha,x,1) computes + bessely(alpha,x).*exp(-abs(imag(x))). + + + + + besselh(alpha [,K] ,x) computes Bessel + functions of the third kind (Hankel function H1 or H2 depending on + K), for real order alpha and + argument x. If omitted K is + supposed to be equal to 1. besselh(alpha,1,x,1) + computes besselh(alpha,1,x).*exp(-%i*x) and + besselh(alpha,2,x,1) computes + besselh(alpha,2,x).*exp(%i*x) + + + + + + Remarks + + If alpha and x are arrays of + the same size, the result y is also that size. If + either input is a scalar, it is expanded to the other input's size. If one + input is a row vector and the other is a column vector, the + resulty is a two-dimensional table of function + values. + + Y_alpha and J_alpha Bessel functions are 2 independent solutions of + the Bessel 's differential equation : + + + + + + + + + K_alpha and I_alpha modified Bessel functions are 2 independant + solutions of the modified Bessel 's differential equation : + + + + + + + + + H^1_alpha and H^2_alpha, the Hankel functions of first and second + kind, are linear linear combinations of Bessel functions of the first and + second kinds: + + + + + + + + + + + Examples + + + + x = linspace(0.01,10,5000)'; + clf() + subplot(2,1,1) + plot2d(x,besseli(0:4,x), style=2:6) + legend('I'+string(0:4),2); + xtitle("Some modified Bessel functions of the first kind") + subplot(2,1,2) + plot2d(x,besseli(0:4,x,1), style=2:6) + legend('I'+string(0:4),1); + xtitle("Some modified scaled Bessel functions of the first kind") + + + + + + + + x = linspace(0,40,5000)'; + plot2d(x,besselj(0:4,x), style=2:6, leg="J0@J1@J2@J3@J4") + legend('I'+string(0:4),1); + xtitle("Some Bessel functions of the first kind") + + + 0 & y2 ~= 0); + clf() + subplot(2,1,1) + plot2d(x,y1,style=2) + xtitle("besselj(0.5,x)") + subplot(2,1,2) + plot2d(x(ind), er(ind), style=2, logflag="nl") + xtitle("relative error between 2 formulae for besselj(0.5,x)") + ]]> + + 0 & y2 ~= 0); + clf() + subplot(2,1,1) + plot2d(x,y1,style=2) + xtitle("besselj(0.5,x)") + subplot(2,1,2) + plot2d(x(ind), er(ind), style=2, logflag="nl") + xtitle("relative error between 2 formulae for besselj(0.5,x)") + ]]> + + + + + x = linspace(0.01,10,5000)'; + clf() + subplot(2,1,1) + plot2d(x,besselk(0:4,x), style=0:4, rect=[0,0,6,10]) + legend('K'+string(0:4),1); + xtitle("Some modified Bessel functions of the second kind") + subplot(2,1,2) + plot2d(x,besselk(0:4,x,1), style=0:4, rect=[0,0,6,10]) + legend('K'+string(0:4),1); + xtitle("Some modified scaled Bessel functions of the second kind") + + + 0+ + clf() + plot2d(x,bessely(0:4,x), style=0:4, rect=[0,-1.5,40,0.6]) + legend('Y'+string(0:4),4); + xtitle("Some Bessel functions of the second kind") + ]]> + + + x = linspace(0.1,40,5000)'; // Y Bessel functions are unbounded for x -> 0+ + clf() + plot2d(x,bessely(0:4,x), style=0:4, rect=[0,-1.5,40,0.6]) + legend('Y'+string(0:4),4); + xtitle("Some Bessel functions of the second kind") + + + + + + x=-4:0.025:2; y=-1.5:0.025:1.5; + [X,Y] = ndgrid(x,y); + H = besselh(0,1,X+%i*Y); + clf();f=gcf(); + xset("fpf"," ") + f.color_map=jetcolormap(16); + contour2d(x,y,abs(H),0.2:0.2:3.2,strf="034",rect=[-4,-1.5,3,1.5]) + legends(string(0.2:0.2:3.2),1:16,"ur") + xtitle("Level curves of |H1(0,z)|") + + + + + Used Functions + The source codes can be found in SCI/modules/special_functions/src/fortran/slatec and + SCI/modules/special_functions/src/fortran + + Slatec : dbesi.f, zbesi.f, dbesj.f, zbesj.f, dbesk.f, zbesk.f, + dbesy.f, zbesy.f, zbesh.f + + Drivers to extend definition area (Serge Steer INRIA): dbesig.f, + zbesig.f, dbesjg.f, zbesjg.f, dbeskg.f, zbeskg.f, dbesyg.f, zbesyg.f, + zbeshg.f + + + diff --git a/modules/special_functions/help/en_US/beta.xml b/modules/special_functions/help/en_US/beta.xml new file mode 100755 index 000000000..2a94b8a80 --- /dev/null +++ b/modules/special_functions/help/en_US/beta.xml @@ -0,0 +1,124 @@ + + + + + beta + beta function (Euler integral of the first kind) + + + Calling Sequence + z = beta(x,y) + + + Arguments + + + x, y + + + 2 positive real scalars, vectors or matricesof equal sizes. + + + + + z + + + a real or a matrix of the same size than x + with z(i,j) = beta(x(i,j),y(i,j)). + + + + + + + Description + Computes the complete beta function : + + + + + + + + + For small x and y (x+y≤2 elementwise), + the algorithm uses the expression in function of the gamma function, else it + applies the exponential function onto the result of the + betaln function provided with the DCDFLIB: Library of + Fortran Routines for Cumulative Distribution Functions, Inverses, and + Other Parameter (see cdfbet for more + information about DCDFLIB). + + + + Examples + + + + x = logspace(-8,8,20000)'; + e = beta(ones(x),x) - (1)./x; + er = abs(e) .* x; + ind = find(er ~= 0); + eps = ones(x(ind))*number_properties("eps"); + plot2d(x(ind),[er(ind) eps 2*eps],style=[1 2 3],logflag="ll",leg="er@eps_m@2 eps_m") + xtitle("approximate relative error in computing beta(1,x)") + + + + t = linspace(0.2,10,60); + X = t'*ones(t); Y = ones(t')*t; + Z = beta(X,Y); + clf() + plot3d(t, t, Z, flag=[2 4 4], leg="x@y@z", alpha=75, theta=30) + xtitle("The beta function on [0.2,10]x[0.2,10]") + + + + See Also + + + gamma + + + cdfbet + + + + diff --git a/modules/special_functions/help/en_US/calerf.xml b/modules/special_functions/help/en_US/calerf.xml new file mode 100755 index 000000000..1ba1dd21f --- /dev/null +++ b/modules/special_functions/help/en_US/calerf.xml @@ -0,0 +1,93 @@ + + + + + calerf + computes error functions. + + + Arguments + + + x + + real vector or matrix + + + + flag + + integer indicator + + + + y + + real vector or matrix (of same size than x) + + + + + + Description + + calerf(x,0) computes the error function + erf(x) + + + calerf(x,1) computes the complementary error + function erfc(x) + + + calerf(x,2) computes the scaled complementary + error function erfcx(x) + + + + Examples + + + + Algorithms + + This function is based on the Faddeeva package library. + + + + See Also + + + erf + + + erfc + + + erfcx + + + + + History + + + 5.5.0 + The calerf function is based on the fadddeeva package + + + + diff --git a/modules/special_functions/help/en_US/dawson.xml b/modules/special_functions/help/en_US/dawson.xml new file mode 100755 index 000000000..4ed3285fd --- /dev/null +++ b/modules/special_functions/help/en_US/dawson.xml @@ -0,0 +1,88 @@ + + + + + dawson + Compute the Dawson (scaled imaginary error) function. + + + Calling Sequence + y = dawson(x) + + + Arguments + + + x + + vector or matrix + + + + y + + vector or matrix (of same size than x) + + + + + + Description + + dawson computes scaled imaginary error function + function: + + + + \mathrm{Dawson}(z) = \frac{\sqrt{\pi}}{2} e^{-z^2} \mathrm{erfi}(z) + + + + + Examples + + + + Algorithms + + This function is based on the Faddeeva package library. + + + + See Also + + + erf + + + erfc + + + calerf + + + + + History + + + 5.5.0 + Function dawson introduced + + + + diff --git a/modules/special_functions/help/en_US/delip.xml b/modules/special_functions/help/en_US/delip.xml new file mode 100755 index 000000000..193e2f39a --- /dev/null +++ b/modules/special_functions/help/en_US/delip.xml @@ -0,0 +1,146 @@ + + + + + delip + complete and incomplete elliptic integral of first + kind + + + + Calling Sequence + [r]=delip(x,ck) + + + Arguments + + + x + + real vector/matrix with nonnegative elements + + + + ck + + real number between -1 and 1 + + + + r + + real or complex number (or vector/matrix) with the same size as + x + + + + + + + Description + The elliptic integral of the first kind with parameter + ck is defined as follow: + + + + + + + + + + ∫ + 0 + x + + + dt + + + + ( + + 1 + − + + t + 2 + + + ) + + + ( + + + 1 + − + + c + k + 2 + + + + t + 2 + + + ) + + + + + + int from{0} to{x} {{dt} over sqrt{ + (1-t^2)(1-c_k^2 t^2)}} + + + + + + + + + Where x is real and positive, + ck is in [-1 1]. + + + If x is less than 1 the result is real. + + + When called with x a vector/matrix r + is evaluated for each entry of x. + + + + Examples + + + + See Also + + + amell + + + %sn + + + + diff --git a/modules/special_functions/help/en_US/dlgamma.xml b/modules/special_functions/help/en_US/dlgamma.xml new file mode 100755 index 000000000..bfade6dd1 --- /dev/null +++ b/modules/special_functions/help/en_US/dlgamma.xml @@ -0,0 +1,84 @@ + + + + + dlgamma + derivative of gammaln function, psi function + + + Calling Sequence + y = dlgamma(x) + + + Arguments + + + x + + real vector + + + + y + + real vector with same size. + + + + + + Description + + dlgamma(x) evaluates, at all the elements of + x the logarithmic derivative of the gamma function + which corresponds also to the derivative of the gammaln function : + + + + + + + + + + x must be real. Also known as the psi + function. + + + + Examples + + + + See Also + + + gamma + + + gammaln + + + + + History + + + 5.4.0 + Overloading allowed for list, mlist, tlist and hypermatrix types. + + + + diff --git a/modules/special_functions/help/en_US/erf.xml b/modules/special_functions/help/en_US/erf.xml new file mode 100755 index 000000000..e456a62bb --- /dev/null +++ b/modules/special_functions/help/en_US/erf.xml @@ -0,0 +1,96 @@ + + + + + erf + The error function. + + + Calling Sequence + y = erf(x) + + + Arguments + + + x + + vector or matrix + + + + y + + vector or matrix (of same size than x) + + + + + + Description + + erf computes the error function: + + \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt + + + + + Examples + + + + Algorithms + + This function is based on the Faddeeva package library. + + + + See Also + + + calerf + + + cdfnor + + + erfc + + + erfcx + + + erfinv + + + + + History + + + 5.5.0 + Function erf supports complex arguments + + + + diff --git a/modules/special_functions/help/en_US/erfc.xml b/modules/special_functions/help/en_US/erfc.xml new file mode 100755 index 000000000..41666de5f --- /dev/null +++ b/modules/special_functions/help/en_US/erfc.xml @@ -0,0 +1,84 @@ + + + + + erfc + The complementary error function. + + + Calling Sequence + y = erfc(x) + + + Arguments + + + x + + vector or matrix + + + + y + + vector or matrix (of same size than x) + + + + + + Description + + Compute the complementary error function of x, defined by: 1- \operatorname{erf}(x) + + + + Examples + + + + Algorithms + + This function is based on the Faddeeva package library. + + + + See Also + + + erf + + + erfcx + + + calerf + + + + + History + + + 5.5.0 + Function erfc supports complex arguments + + + + diff --git a/modules/special_functions/help/en_US/erfcx.xml b/modules/special_functions/help/en_US/erfcx.xml new file mode 100755 index 000000000..0a85ce2f2 --- /dev/null +++ b/modules/special_functions/help/en_US/erfcx.xml @@ -0,0 +1,85 @@ + + + + + erfcx + scaled complementary error function. + + + Calling Sequence + y = erfcx(x) + + + Arguments + + + x + + vector or matrix + + + + y + + vector or matrix (of same size than x) + + + + + + Description + + Compute the scaled complementary error function of x, defined by e^{x^2} \operatorname{erfc}(x). Note also that \operatorname{erfcx}(-ix) computes the Faddeeva function w(x). + + + + + Examples + + + + Algorithms + + This function is based on the Faddeeva package library. + + + + See Also + + + erf + + + erfc + + + calerf + + + + + History + + + 5.5.0 + Improve the behavior of erfcx on big values. + + + + + diff --git a/modules/special_functions/help/en_US/erfi.xml b/modules/special_functions/help/en_US/erfi.xml new file mode 100755 index 000000000..93ba46c11 --- /dev/null +++ b/modules/special_functions/help/en_US/erfi.xml @@ -0,0 +1,88 @@ + + + + + erfi + The imaginary error function. + + + Calling Sequence + y = erfi(z) + + + Arguments + + + z + + vector or matrix + + + + y + + vector or matrix (of same size than x) + + + + + + Description + + erfi computes the imaginary error function of x, defined by -i \operatorname{erf}(ix) + + + + Examples + + + + Algorithms + + This function is based on the Faddeeva package library. + + + + See Also + + + erf + + + erfcx + + + erfc + + + calerf + + + + + History + + + 5.5.0 + Function erfi supports complex arguments + + + + diff --git a/modules/special_functions/help/en_US/erfinv.xml b/modules/special_functions/help/en_US/erfinv.xml new file mode 100755 index 000000000..f511219b6 --- /dev/null +++ b/modules/special_functions/help/en_US/erfinv.xml @@ -0,0 +1,89 @@ + + + + + erfinv + inverse error function + + + Calling Sequence + y = erfinv(x) + + + Arguments + + + x + + vector or matrix + + + + y + + vector or matrix (same size as x) + + + + + + Description + + The erfinv function computes the inverse of + the erf error function. Thus, erf(erfinv(x)) + = x for all x such that -1 ≤ x ≤ + 1 + + . + + + + Examples + + + x = linspace(-0.99, 0.99, 100); + y = erfinv(x); + plot2d(x, y); + + + + See Also + + + cdfnor + + + erf + + + erfc + + + erfcx + + + + + References + + Milton Abramowitz and Irene A. Stegun, eds. Handbook of + Mathematical Functions with Formulas, Graphs, and Mathematical + Tables. New York: Dover, 1972. + + + diff --git a/modules/special_functions/help/en_US/findm.xml b/modules/special_functions/help/en_US/findm.xml new file mode 100755 index 000000000..af1f5f526 --- /dev/null +++ b/modules/special_functions/help/en_US/findm.xml @@ -0,0 +1,26 @@ + + + + findm + for elliptic filter design + + + Calling Sequence + [m]=findm(chi) + + + Description + + Search for m such that chi = %k(1-m)/%k(m) + (For use with find_freq). + + + + See Also + + + %k + + + + diff --git a/modules/special_functions/help/en_US/gamma.xml b/modules/special_functions/help/en_US/gamma.xml new file mode 100755 index 000000000..464c05a2d --- /dev/null +++ b/modules/special_functions/help/en_US/gamma.xml @@ -0,0 +1,105 @@ + + + + + gamma + The gamma function. + + + Calling Sequence + y = gamma(x) + + + Arguments + + + x + + real vector or matrix + + + + y + + real vector or matrix with same size than x. + + + + + + Description + + gamma(x) evaluates the gamma function at all the + elements of x. The gamma function is defined by + : + + + + + + + + + and generalizes the factorial function for real numbers + (gamma(n+1) = n!). + + + + Examples + + + + a = -3; b = 5; + x = linspace(a,b,40000)'; + y = gamma(x); + plot2d(x, y, style=0, axesflag=5, rect=[a, -10, b, 10]) + xtitle("The gamma function on ["+string(a)+","+string(b)+"]") + + + + See Also + + + gammaln + + + dlgamma + + + + + History + + + 5.4.0 + Overloading allowed for list, mlist, tlist and hypermatrix types. + + + + diff --git a/modules/special_functions/help/en_US/gammaln.xml b/modules/special_functions/help/en_US/gammaln.xml new file mode 100755 index 000000000..b66942ae8 --- /dev/null +++ b/modules/special_functions/help/en_US/gammaln.xml @@ -0,0 +1,73 @@ + + + + + gammaln + The logarithm of gamma function. + + + Calling Sequence + y = gammaln(x) + + + Arguments + + + x + + real vector + + + + y + + real vector with same size. + + + + + + Description + + gammaln(x) evaluates the logarithm of gamma + function at all the elements of x, avoiding underflow + and overflow. x must be real. + + + + Examples + + + + See Also + + + gamma + + + dlgamma + + + + + History + + + 5.4.0 + Overloading allowed for list, mlist, tlist and hypermatrix types. + + + + diff --git a/modules/special_functions/help/en_US/legendre.xml b/modules/special_functions/help/en_US/legendre.xml new file mode 100755 index 000000000..db287e60c --- /dev/null +++ b/modules/special_functions/help/en_US/legendre.xml @@ -0,0 +1,230 @@ + + + + + legendre + associated Legendre functions + + + Calling Sequence + y = legendre(n,m,x [,normflag]) + + + Arguments + + + n + + non negative integer or vector of non negative integers + regularly spaced with increment equal to 1 + + + + + m + + non negative integer or vector of non negative integers + regularly spaced with increment equal to 1 + + + + + x + + + real matrix(elements of x must be in + the [-1,1] interval) + + + + + normflag + + (optional) scalar string + + + + + + Description + + When n and m are scalars, + legendre(n,m,x) evaluates the associated Legendre + function Pnm(x) at all the elements of x. The + definition used is : + + + + + + + + + + where Pn is the Legendre polynomial of degree + n. So legendre(n,0,x) evaluates the + Legendre polynomial Pn(x) at all the elements of + x. + + When the normflag is equal to "norm" you get a normalized version + (without the (-1)^m factor), precisely : + + + + + + + + + which is useful to compute spherical harmonic functions (see Example + 3): + + For efficiency, one of the two first arguments may be a vector, for + instance legendre(n1:n2,0,x) evaluates all the Legendre + polynomials of degree n1, n1+1, ..., n2 at the + elements of x and + legendre(n,m1:m2,x) evaluates all the Legendre + associated functions Pnm for m=m1, m1+1, ..., m2 at + x. + + + + Output format + + In any case, the format of y is : + + + and : + + + so that x is preferably a row vector but any + mx x nx matrix is expected and considered as an + 1 x (mx * nx) matrix, reshaped following the column + order. + + + + Examples + + + l = nearfloat("pred",1); + x = linspace(-l,l,200)'; + y = legendre(0:5, 0, x); + plot2d(x,y', leg="p0@p1@p2@p3@p4@p5@p6") + xtitle("the 6 th first Legendre polynomials") + + + + l = nearfloat("pred",1); + x = linspace(-l,l,200)'; + y = legendre(5, 0:5, x, "norm"); + plot2d(x,y', leg="p5,0@p5,1@p5,2@p5,3@p5,4@p5,5") + xtitle("the (normalized) associated Legendre functions of degree 5") + + = 0 then + y = (-1)^m/(sqrt(2*%pi))*exp(%i*m*phi)*legendre(l, m, cos(theta), "norm") + else + y = 1/(sqrt(2*%pi))*exp(%i*m*phi)*legendre(l, -m, cos(theta), "norm") + end +endfunction + +// 3.2 : define another useful function +function [x,y,z] = sph2cart(theta,phi,r) + // theta row vector 1 x nt + // phi column vector np x 1 + // r scalar or np x nt matrix (r(i,j) the length at phi(i) theta(j)) + x = r.*(cos(phi)*sin(theta)); + y = r.*(sin(phi)*sin(theta)); + z = r.*(ones(phi)*cos(theta)); +endfunction + +// 3-3 plot Y31(theta,phi) +l = 3; m = 1; +theta = linspace(0.1,%pi-0.1,60); +phi = linspace(0,2*%pi,120)'; +f = Y(l,m,theta,phi); +[x1,y1,z1] = sph2cart(theta,phi,abs(f)); [xf1,yf1,zf1] = nf3d(x1,y1,z1); +[x2,y2,z2] = sph2cart(theta,phi,abs(real(f))); [xf2,yf2,zf2] = nf3d(x2,y2,z2); +[x3,y3,z3] = sph2cart(theta,phi,abs(imag(f))); [xf3,yf3,zf3] = nf3d(x3,y3,z3); + +clf() +subplot(1,3,1) +plot3d(xf1,yf1,zf1,flag=[2 4 4]); xtitle("|Y31(theta,phi)|") +subplot(1,3,2) +plot3d(xf2,yf2,zf2,flag=[2 4 4]); xtitle("|Real(Y31(theta,phi))|") +subplot(1,3,3) +plot3d(xf3,yf3,zf3,flag=[2 4 4]); xtitle("|Imag(Y31(theta,phi))|") + ]]> + + + diff --git a/modules/special_functions/help/en_US/percentk.xml b/modules/special_functions/help/en_US/percentk.xml new file mode 100755 index 000000000..a57a3d821 --- /dev/null +++ b/modules/special_functions/help/en_US/percentk.xml @@ -0,0 +1,59 @@ + + + + %k + Jacobi's complete elliptic integral + + + Calling Sequence + [K]=%k(m) + + + Arguments + + + m + + + parameter of the elliptic integral 0<m<1 (m can be a vector) + + + + + K + + value of the elliptic integral from 0 to 1 on the real axis + + + + + + Description + + Calculates Jacobi's complete elliptic integral + of the first kind : + + + + References + + Abramowitz and Stegun page 598 + + + + Examples + + + + See Also + + + delip + + + + diff --git a/modules/special_functions/help/en_US/percentsn.xml b/modules/special_functions/help/en_US/percentsn.xml new file mode 100755 index 000000000..6e5fc66f9 --- /dev/null +++ b/modules/special_functions/help/en_US/percentsn.xml @@ -0,0 +1,100 @@ + + + + %sn + Jacobi's elliptic function + + + Calling Sequence + [y]=%sn(x,m) + + + Arguments + + + x + + + a point inside the fundamental rectangle defined by the elliptic integral; x is a vector of complex numbers + + + + + m + + + parameter of the elliptic integral (0<m<1) + + + + + y + + result + + + + + + Description + + Jacobi 's sn elliptic function with parameter m: the inverse + of the elliptic integral for the parameter m. + + + The amplitude am is computed in fortran and + the addition formulas for elliptic functions are applied + + + + Examples + + + m=0.36; + K=%k(m); + P=4*K; + real_val=0:(P/50):P; + plot(real_val,real(%sn(real_val,m))) + + + + m=0.36; + KK=%k(1-m); + Ip=2*KK; + ima_val1=0:(Ip/50):KK-0.001; + ima_val2=(KK+0.05):(Ip/25):(Ip+KK); + z1=%sn(%i*ima_val1,m); + z2=%sn(%i*ima_val2,m); + plot2d([ima_val1',ima_val2'],[imag(z1)',imag(z2)']); + xgrid(3) + + + + See Also + + + delip + + + %k + + + + diff --git a/modules/special_functions/help/fr_FR/addchapter.sce b/modules/special_functions/help/fr_FR/addchapter.sce new file mode 100755 index 000000000..51491de7d --- /dev/null +++ b/modules/special_functions/help/fr_FR/addchapter.sce @@ -0,0 +1,11 @@ +// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +// Copyright (C) 2009 - DIGITEO +// +// This file must be used under the terms of the CeCILL. +// This source file is licensed as described in the file COPYING, which +// you should have received as part of this distribution. The terms +// are also available at +// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt + +add_help_chapter("Fonctions spéciales",SCI+"/modules/special_functions/help/fr_FR",%T); + diff --git a/modules/special_functions/help/fr_FR/amell.xml b/modules/special_functions/help/fr_FR/amell.xml new file mode 100755 index 000000000..5dda0c197 --- /dev/null +++ b/modules/special_functions/help/fr_FR/amell.xml @@ -0,0 +1,57 @@ + + + + amell + Fonction am de Jacobi + + + Séquence d'appel + [sn]=amell(u,k) + + + Paramètres + + + u + + scalaire ou vecteur réel + + + + + k + + scalaire + + + + + sn + + scalaire ou vecteur réel + + + + + + + Description + + Calcule la fonction elliptique de Jacobi am(u,k) + où k est le paramètre et u l'argument. Si u + est un vecteur sn est le vecteur des valeurs calculées (élément par élément). + Utilisé dans la fonction %sn. + + + + Voir aussi + + + delip + + + %sn + + + + diff --git a/modules/special_functions/help/fr_FR/calerf.xml b/modules/special_functions/help/fr_FR/calerf.xml new file mode 100755 index 000000000..80c729be0 --- /dev/null +++ b/modules/special_functions/help/fr_FR/calerf.xml @@ -0,0 +1,78 @@ + + + + + calerf + calcule différentes fonctions d'erreur. + + + Paramètres + + + x + + vecteur réel + + + + + flag + + un entier + + + + + y + + vecteur réel (de même taille que x) + + + + + + + Description + + calerf(x,0) calcule la fonction erreur :erf(x) + + + calerf(x,1) calcule la fonction erreur complémentaire :erfc(x) + + + calerf(x,2) calcule la fonction erreur complémentaire normalisée :erfcx(x) + + + + Exemples + + + + Voir aussi + + + erf + + + erfc + + + calerf + + + + diff --git a/modules/special_functions/help/fr_FR/delip.xml b/modules/special_functions/help/fr_FR/delip.xml new file mode 100755 index 000000000..d573d8c2b --- /dev/null +++ b/modules/special_functions/help/fr_FR/delip.xml @@ -0,0 +1,134 @@ + + + + delip + intégrale elliptique complete ou incomplete du premier + type + + + + Séquence d'appel + [r]=delip(x,ck) + + + Paramètres + + + x + + vecteur réel à éléments non négatifs + + + + ck + + scalaire entre -1 et 1 + + + + r + + réel ou complexe (ou vecteur) + + + + + + Description + L'integrale élliptique du premier type pour le paramètre + ck est définie par: + + + + + + + + + + ∫ + 0 + x + + + dt + + + + ( + + 1 + − + + t + 2 + + + ) + + + ( + + + 1 + − + + c + k + 2 + + + + t + 2 + + + ) + + + + + + int from{0} to{x} {{dt} over sqrt{ + (1-t^2)(1-c_k^2 t^2)}} + + + + + + + + + Où x est réel et positif,ck + est dans l'intervalle [-1 1]. + + + Si x est plus petit ou égal a 1 le resultat est + réel + + + Quand x est un vecteur r est + évalué pour chaque composante de x. + + + + Exemples + + + + Voir aussi + + + amell + + + %sn + + + + diff --git a/modules/special_functions/help/fr_FR/dlgamma.xml b/modules/special_functions/help/fr_FR/dlgamma.xml new file mode 100755 index 000000000..5e1ba47ff --- /dev/null +++ b/modules/special_functions/help/fr_FR/dlgamma.xml @@ -0,0 +1,83 @@ + + + + + dlgamma + dérivée de la fonction gammaln ou fonction psi. + + + Séquence d'appel + y = dlgamma(x) + + + Paramètres + + + x + + vecteur réel. + + + + y + + vecteur réel de même taille. + + + + + + Description + + dlgamma(x) calcule la dérivée de la fonction + gammaln pour chaque composante de x. + + + + + + + + + + x doit être réel. Cette fonction est aussi connue + sous le nom de fonction psi + + + + Exemples + + + + Voir aussi + + + gamma + + + gammaln + + + + + Historique + + + 5.4.0 + Surcharge autorisée pour les types list, mlist, tlist et hypermatrices. + + + + diff --git a/modules/special_functions/help/fr_FR/erf.xml b/modules/special_functions/help/fr_FR/erf.xml new file mode 100755 index 000000000..7b9326660 --- /dev/null +++ b/modules/special_functions/help/fr_FR/erf.xml @@ -0,0 +1,88 @@ + + + + erf + fonction erreur + + + Séquence d'appel + y = erf(x) + + + Paramètres + + + x + + vecteur + + + + y + + vecteur (de même taille que x) + + + + + + Description + + erfcalcule la fonction erreur : + + \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt + + + + + Exemples + + + + + Algorithmes + + Cette fonction est basée sur la bibliothèque Faddeeva. + + + + + Voir aussi + + + calerf + + + cdfnor + + + erfc + + + erfcx + + + erfinv + + + + + + Historique + + + 5.5.0 + La fonction erf supporte les arguments complexes. + + + + diff --git a/modules/special_functions/help/fr_FR/erfc.xml b/modules/special_functions/help/fr_FR/erfc.xml new file mode 100755 index 000000000..ac1927892 --- /dev/null +++ b/modules/special_functions/help/fr_FR/erfc.xml @@ -0,0 +1,74 @@ + + + + erfc + fonction erreur complémentaire. + + + Séquence d'appel + y = erfc(x) + + + Paramètres + + + x + + vecteur + + + + y + + vecteur (de même taille que x) + + + + + + Description + + erfc calcule la fonction erreur + complémentaire définie par : 1- \operatorname{erf}(x) + + + + Exemples + + + + + Algorithmes + + Cette fonction est basée sur la bibliothèque Faddeeva. + + + + Voir aussi + + + erf + + + erfcx + + + calerf + + + + + Historique + + + 5.5.0 + La fonction erfc supporte les arguments complexes. + + + + diff --git a/modules/special_functions/help/fr_FR/erfcx.xml b/modules/special_functions/help/fr_FR/erfcx.xml new file mode 100755 index 000000000..f138774a0 --- /dev/null +++ b/modules/special_functions/help/fr_FR/erfcx.xml @@ -0,0 +1,75 @@ + + + + erfcx + fonction erreur complémentaire normalisée. + + + Séquence d'appel + y = erfcx(x) + + + Paramètres + + + x + + vecteur + + + + y + + vecteur (de même taille que x) + + + + + + Description + + erfcx calcule la fonction erreur complémentaire + normalisée définie par e^{x^2} \operatorname{erfc}(x). \operatorname{erfcx}(-ix) calcul aussi la fonction Faddeeva w(x). + + + + Exemples + + + + + Algorithmes + + Cette fonction est basée sur la bibliothèque Faddeeva. + + + + Voir aussi + + + erf + + + erfc + + + calerf + + + + + + Historique + + + 5.5.0 + La fonction erfc supporte les arguments complexes. + + + + diff --git a/modules/special_functions/help/fr_FR/erfinv.xml b/modules/special_functions/help/fr_FR/erfinv.xml new file mode 100755 index 000000000..17e4c86b8 --- /dev/null +++ b/modules/special_functions/help/fr_FR/erfinv.xml @@ -0,0 +1,80 @@ + + + + + erfinv + fonction erreur inverse + + + Séquence d'appel + y = erfinv(x) + + + Paramètres + + + x + + vecteur réel + + + + y + + vecteur réel (de même taille que x) + + + + + + Description + + La fonction erfinv calcule l'inverse de la + fonction d'erreur erf. Donc, erf(erfinv(x)) = + x pour tout x tel que -1 ≤ x ≤ + 1 + + . + + + + Exemples + + + + Voir aussi + + + cdfnor + + + erf + + + erfc + + + erfcx + + + + + Références + Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. + + diff --git a/modules/special_functions/help/fr_FR/gammaln.xml b/modules/special_functions/help/fr_FR/gammaln.xml new file mode 100755 index 000000000..713cc3745 --- /dev/null +++ b/modules/special_functions/help/fr_FR/gammaln.xml @@ -0,0 +1,76 @@ + + + + + gammaln + Le logarithme de la fonction gamma. + + + Séquence d'appel + y = gammaln(x) + + + Paramètres + + + x + + vecteur ou matrice de nombres réels. + + + + y + + vecteur ou matrice de nombres réels de même taille que + x. + + + + + + + Description + + gammaln(x) évalue le logarithme de la fonction + gamma pour les composantes de x., en évitant les + valeurs conduisant à un underflow ou un overflow. x + doit être à composantes réelles. + + + + Exemples + + + + Voir aussi + + + gamma + + + dlgamma + + + + + Historique + + + 5.4.0 + Surcharge autorisée pour les types list, mlist, tlist et hypermatrices. + + + + diff --git a/modules/special_functions/help/ja_JP/addchapter.sce b/modules/special_functions/help/ja_JP/addchapter.sce new file mode 100755 index 000000000..99079eca7 --- /dev/null +++ b/modules/special_functions/help/ja_JP/addchapter.sce @@ -0,0 +1,11 @@ +// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +// Copyright (C) 2009 - DIGITEO +// +// This file must be used under the terms of the CeCILL. +// This source file is licensed as described in the file COPYING, which +// you should have received as part of this distribution. The terms +// are also available at +// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt + +add_help_chapter("Special Functions",SCI+"/modules/special_functions/help/ja_JP",%T); + diff --git a/modules/special_functions/help/ja_JP/amell.xml b/modules/special_functions/help/ja_JP/amell.xml new file mode 100755 index 000000000..a04d96250 --- /dev/null +++ b/modules/special_functions/help/ja_JP/amell.xml @@ -0,0 +1,66 @@ + + + + + amell + ヤコビのam関数 + + + 呼び出し手順 + [sn]=amell(u,k) + + + パラメータ + + + u + + 実数スカラーまたはベクトル + + + + k + + スカラー + + + + sn + + 実数スカラーまたはベクトル + + + + + + 説明 + + ヤコビの楕円関数 am(u,k)を計算します. + ただし, + k はパラメータ, u は引数です. u + はベクトル, sn は(要素毎の)計算値のベクトルです. + 関数%snで使用されます. + + + + 参照 + + + delip + + + %sn + + + + diff --git a/modules/special_functions/help/ja_JP/bessel.xml b/modules/special_functions/help/ja_JP/bessel.xml new file mode 100755 index 000000000..10598930e --- /dev/null +++ b/modules/special_functions/help/ja_JP/bessel.xml @@ -0,0 +1,322 @@ + + + + + besseli + 第1種修正ベッセル関数 (I_alpha). + + + besselj + 第1種ベッセル関数 (J_alpha). + + + besselk + 第2種修正ベッセル関数 (K_alpha). + + + bessely + 第2種ベッセル関数 (Y_alpha). + + + besselh + 第3種ベッセル関数 (ハンケル関数と同じ) + + + 呼び出し手順 + y = besseli(alpha,x [,ice]) + y = besselj(alpha,x [,ice]) + y = besselk(alpha,x [,ice]) + y = bessely(alpha,x [,ice]) + y = besselh(alpha,x) + y = besselh(alpha,K,x [,ice]) + + + + 引数 + + + x + + 実数または複素数のベクトル. + + + + alpha + + r実数ベクトル + + + + ice + + 整数フラグ, デフォルト値: 0 + + + + K + + 整数, 指定可能な値は 1 または 2, ハンケル関数の型. + + + + + + 説明 + + + + besseli(alpha,x) は, + 実数の次数alpha および引数 xに関する + 第1種修正ベッセル関数(I_alpha)を計算します, + besseli(alpha,x,1) は + besseli(alpha,x).*exp(-abs(real(x)))を計算します. + + + + + besselj(alpha,x) は第1種のベッセル関数(J_alpha)を + 実数の次数alpha および引数 xに関して + 計算します. + besselj(alpha,x,1) は + besselj(alpha,x).*exp(-abs(imag(x)))を計算します. + + + + + besselk(alpha,x) は第2種修正ベッセル関数 + (K_alpha)を + 実数の次数alpha および引数 xに関して + 計算します. + besselk(alpha,x,1) は + besselk(alpha,x).*exp(x)を計算します. + + + + + bessely(alpha,x)は第2種のベッセル関数(Y_alpha)を + 実数の次数alpha および引数 xに関して + 計算します. + bessely(alpha,x,1) は + bessely(alpha,x).*exp(-abs(imag(x)))を計算します. + + + + + besselh(alpha [,K] ,x) は第3種のベッセル関数 + (Kに依存してハンケル関数 H1 または H2)を + 実数の次数alpha および引数 xに関して + 計算します.Kが省略された場合, + 1に等しいと仮定されます. + besselh(alpha,1,x,1)は + besselh(alpha,1,x).*exp(-%i*x)を計算し, + besselh(alpha,2,x,1) は + besselh(alpha,2,x).*exp(%i*x)を計算します. + + + + + + 注意 + + alphaおよび xが同じ大きさの + 配列の場合,結果yも同じ大きさとなります. + 入力のどちらかがスカラーの場合, + もう片方の大きさにまで拡張されます. + 片方の入力が行ベクトルでもう片方が列ベクトルの場合, + 結果yは関数値の二次元テーブルとなります. + + Y_alpha および J_alpha ベッセル関数はベッセルの微分方程式の + 2つの独立解です: + + + + + + + + + 修正ベッセル関数K_alpha および I_alphaは + 修正ベッセル微分方程式の2つの独立解です: + + + + + + + + + H^1_alpha および H^2_alphaは第1種および第2種のハンケル関数 + で,第1種および第2種のベッセル関数の線形結合です: + + + + + + + + + + + 例 + + + x = linspace(0.01,10,5000)'; + clf() + subplot(2,1,1) + plot2d(x,besseli(0:4,x), style=2:6) + legend('I'+string(0:4),2); + xtitle("Some modified Bessel functions of the first kind") + subplot(2,1,2) + plot2d(x,besseli(0:4,x,1), style=2:6) + legend('I'+string(0:4),1); + xtitle("Some modified scaled Bessel functions of the first kind") + + + + x = linspace(0,40,5000)'; + plot2d(x,besselj(0:4,x), style=2:6, leg="J0@J1@J2@J3@J4") + legend('I'+string(0:4),1); + xtitle("Some Bessel functions of the first kind") + + + 0 & y2 ~= 0); + clf() + subplot(2,1,1) + plot2d(x,y1,style=2) + xtitle("besselj(0.5,x)") + subplot(2,1,2) + plot2d(x(ind), er(ind), style=2, logflag="nl") + xtitle("relative error between 2 formulae for besselj(0.5,x)") + ]]> + + + x = linspace(0.01,10,5000)'; + clf() + subplot(2,1,1) + plot2d(x,besselk(0:4,x), style=0:4, rect=[0,0,6,10]) + legend('K'+string(0:4),1); + xtitle("Some modified Bessel functions of the second kind") + subplot(2,1,2) + plot2d(x,besselk(0:4,x,1), style=0:4, rect=[0,0,6,10]) + legend('K'+string(0:4),1); + xtitle("Some modified scaled Bessel functions of the second kind") + + + + x = linspace(0.1,40,5000)'; // Y Bessel functions are unbounded for x -> 0+ + clf() + plot2d(x,bessely(0:4,x), style=0:4, rect=[0,-1.5,40,0.6]) + legend('Y'+string(0:4),4); + xtitle("Some Bessel functions of the second kind") + + + + x=-4:0.025:2; y=-1.5:0.025:1.5; + [X,Y] = ndgrid(x,y); + H = besselh(0,1,X+%i*Y); + clf();f=gcf(); + xset("fpf"," ") + f.color_map=jetcolormap(16); + contour2d(x,y,abs(H),0.2:0.2:3.2,strf="034",rect=[-4,-1.5,3,1.5]) + legends(string(0.2:0.2:3.2),1:16,"ur") + xtitle("Level curves of |H1(0,z)|") + + + + 使用される関数 + ソースコードは SCI/modules/special_functions/src/fortran/slatec および + SCI/modules/special_functions/src/fortran にあります + + Slatec : dbesi.f, zbesi.f, dbesj.f, zbesj.f, dbesk.f, zbesk.f, + dbesy.f, zbesy.f, zbesh.f + + 拡張定義領域(Serge Steer INRIA)のドライバ (Serge Steer INRIA): dbesig.f, + zbesig.f, dbesjg.f, zbesjg.f, dbeskg.f, zbeskg.f, dbesyg.f, zbesyg.f, + zbeshg.f + + + diff --git a/modules/special_functions/help/ja_JP/beta.xml b/modules/special_functions/help/ja_JP/beta.xml new file mode 100755 index 000000000..5453c2a36 --- /dev/null +++ b/modules/special_functions/help/ja_JP/beta.xml @@ -0,0 +1,126 @@ + + + + + beta + ベータ関数 (第１種オイラー積分) + + + 呼び出し手順 + z = beta(x,y) + + + パラメータ + + + x, y + + + 正の実数または同じ大きさの正の実数の行列(ベクトル). + + + + + z + + + 実数または + z(i,j) = beta(x(i,j),y(i,j))となる + xと同じ大きさの行列. + + + + + + + 説明 + 完全ベータ関数を計算します : + + + + + + + + + x およびyが小さい場合, + このアルゴリズムは,関数内でガンマ関数の式を使用します. + そうでない場合はDCDFLIBにより提供される + betaln関数の結果に指数関数を適用します: + DCDFLIBは累積密度関数,逆,およびその他のパラメータに関する + Fortranルーチンのライブラリです + (DCDFLIBに関する詳細については cdfbet を参照 + ). + + + + 例 + + + + x = logspace(-8,8,20000)'; + e = beta(ones(x),x) - (1)./x; + er = abs(e) .* x; + ind = find(er ~= 0); + eps = ones(x(ind))*number_properties("eps"); + plot2d(x(ind),[er(ind) eps 2*eps],style=[1 2 3],logflag="ll",leg="er@eps_m@2 eps_m") + xtitle("approximate relative error in computing beta(1,x)") + + + + t = linspace(0.2,10,60); + X = t'*ones(t); Y = ones(t')*t; + Z = beta(X,Y); + clf() + plot3d(t, t, Z, flag=[2 4 4], leg="x@y@z", alpha=75, theta=30) + xtitle("The beta function on [0.2,10]x[0.2,10]") + + + + 参照 + + + gamma + + + cdfbet + + + + diff --git a/modules/special_functions/help/ja_JP/calerf.xml b/modules/special_functions/help/ja_JP/calerf.xml new file mode 100755 index 000000000..c0a9f8e1b --- /dev/null +++ b/modules/special_functions/help/ja_JP/calerf.xml @@ -0,0 +1,94 @@ + + + + + calerf + 誤差関数を計算する. + + + 引数 + + + x + + 実数ベクトルまたは行列 + + + + flag + + 整数インジケータ + + + + y + + (xと同じ大きさの)実数ベクトルまたは行列 + + + + + + 説明 + + calerf(x,0) は誤差関数 + erf(x)を計算します + + + calerf(x,1) は相補誤差関数 + erfc(x)を計算します + + + calerf(x,2) はスケーリング付き相補誤差関数 + erfcx(x)を計算します + + + + 例 + + + + アルゴリズム + + この関数はFaddeevaパッケージライブラリ + に基づきます. + + + + 参照 + + + erf + + + erfc + + + erfcx + + + + + 履歴 + + + 5.5.0 + calerf関数はfadddeevaパッケージを使用しています + + + + diff --git a/modules/special_functions/help/ja_JP/dawson.xml b/modules/special_functions/help/ja_JP/dawson.xml new file mode 100755 index 000000000..8204c1ae2 --- /dev/null +++ b/modules/special_functions/help/ja_JP/dawson.xml @@ -0,0 +1,87 @@ + + + + + dawson + Dawson (スケーリングされた虚数の誤差) 関数を計算. + + + 呼び出し手順 + y = dawson(x) + + + 引数 + + + x + + ベクトルまたは行列 + + + + y + + (xと同じ大きさ)ベクトルまたは行列 + + + + + + 説明 + + dawson はスケーリングされた虚数の誤差関数を計算します: + + + + \mathrm{Dawson}(z) = \frac{\sqrt{\pi}}{2} e^{-z^2} \mathrm{erfi}(z) + + + + + 例 + + + + アルゴリズム + + この関数は Faddeevaパッケージ ライブラリにもとづいています. + + + + 参照 + + + erf + + + erfc + + + calerf + + + + + 履歴 + + + 5.5.0 + 関数dawsonが追加されました + + + + diff --git a/modules/special_functions/help/ja_JP/delip.xml b/modules/special_functions/help/ja_JP/delip.xml new file mode 100755 index 000000000..ddecaa243 --- /dev/null +++ b/modules/special_functions/help/ja_JP/delip.xml @@ -0,0 +1,150 @@ + + + + + delip + + 第一種の完全および不完全楕円積分 + + + + 呼び出し手順 + [r]=delip(x,ck) + + + パラメータ + + + x + + 非負の要素を有する実数ベクトル + + + + ck + + -1 と 1の間の実数 + + + + r + + + xと同じ大きさの + 実数または複素数(またはベクトル) + + + + + + + 説明 + + 引数ckの + 第一種の楕円積分は以下のように定義されます: + + + + + + + + + + ∫ + 0 + x + + + dt + + + + ( + + 1 + − + + t + 2 + + + ) + + + ( + + + 1 + − + + c + k + 2 + + + + t + 2 + + + ) + + + + + + int from{0} to{x} {{dt} over sqrt{ + (1-t^2)(1-c_k^2 t^2)}} + + + + + + + + + ただし, x は実数および正で, + ck は [-1 1]の範囲となります. + + x が1より小さい場合,結果は実数となります. + + xを指定して + コールされた場合,xの各エントリについて + ベクトルrが評価されます. + + + + 例 + + + + 参照 + + + amell + + + delip + + + %sn + + + + diff --git a/modules/special_functions/help/ja_JP/dlgamma.xml b/modules/special_functions/help/ja_JP/dlgamma.xml new file mode 100755 index 000000000..4cfb3fc81 --- /dev/null +++ b/modules/special_functions/help/ja_JP/dlgamma.xml @@ -0,0 +1,86 @@ + + + + +dlgamma +ガンマ関数, psi関数の微分 + + +呼び出し手順 +y = dlgamma(x) + + +引数 + + + x + + 実数ベクトル + + + + y + + 同じ大きさの実数ベクトル. + + + + + +説明 + + dlgamma(x) は,x + の全ての要素について,ガンマ関数の対数微分を計算します. + この値はgammaln関数の微分に一致します: + + + + + + + + + + x は実数である必要があります. + psi 関数としても知られています. + + + +例 + + + +参照 + + + gamma + + + gammaln + + + + +履歴 + + + 5.4.0 + list, mlist, tlist およびハイパー行列型の + オーバーロードが可能となりました. + + + + + diff --git a/modules/special_functions/help/ja_JP/erf.xml b/modules/special_functions/help/ja_JP/erf.xml new file mode 100755 index 000000000..c2280daac --- /dev/null +++ b/modules/special_functions/help/ja_JP/erf.xml @@ -0,0 +1,94 @@ + + + + + erf + 誤差関数. + + + 呼び出し手順 + y = erf(x) + + + 引数 + + + x + + ベクトルまたは行列 + + + + y + + (xと同じ大きさの)ベクトルまたは行列 + + + + + + 説明 + + erf は誤差関数を計算します: + \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt + + + + + 例 + + + + アルゴリズム + + この関数は Faddeevaパッケージ + ライブラリに基づきます. + + + + 参照 + + + calerf + + + cdfnor + + + erfc + + + erfcx + + + erfinv + + + + + 履歴 + + + 5.5.0 + erf関数が複素数の要素をサポート + + + + diff --git a/modules/special_functions/help/ja_JP/erfc.xml b/modules/special_functions/help/ja_JP/erfc.xml new file mode 100755 index 000000000..a36e33bfb --- /dev/null +++ b/modules/special_functions/help/ja_JP/erfc.xml @@ -0,0 +1,83 @@ + + + + + erfc + 相補誤差関数. + + + 呼び出し手順 + y = erfc(x) + + + パラメータ + + + x + + ベクトルまたは行列 + + + + y + + (xと同じ大きさの)ベクトルまたは行列 + + + + + + 説明 + + erfc は相補誤差関数を計算します: 1- \operatorname{erf}(x) + + + + 例 + + + + アルゴリズム + + この関数は Faddeevaパッケージ + ライブラリに基づきます. + + + + 参照 + + + erf + + + erfcx + + + calerf + + + + + 履歴 + + + 5.5.0 + erfc関数が複素数の要素をサポート + + + + diff --git a/modules/special_functions/help/ja_JP/erfcx.xml b/modules/special_functions/help/ja_JP/erfcx.xml new file mode 100755 index 000000000..1f3dcaa76 --- /dev/null +++ b/modules/special_functions/help/ja_JP/erfcx.xml @@ -0,0 +1,84 @@ + + + + + erfcx + スケーリング付き相補誤差関数. + + + 呼び出し手順 + y = erfcx(x) + + + 引数 + + + x + + ベクトルまたは行列 + + + + y + + (xと同じ大きさの)ベクトルまたは行列 + + + + + + 説明 + + erfcx はスケーリング付き相互関数を計算します:e^{x^2} \operatorname{erfc}(x) + + + + 例 + + + + アルゴリズム + + この関数は Faddeevaパッケージ + ライブラリに基づきます. + + + + 参照 + + + erf + + + erfc + + + calerf + + + + + 履歴 + + + 5.5.0 + erfcx関数が複素数の要素をサポート + + + + diff --git a/modules/special_functions/help/ja_JP/erfi.xml b/modules/special_functions/help/ja_JP/erfi.xml new file mode 100755 index 000000000..b7d89a505 --- /dev/null +++ b/modules/special_functions/help/ja_JP/erfi.xml @@ -0,0 +1,88 @@ + + + + + erfi + 虚数の誤差関数. + + + 呼び出し手順 + y = erfi(z) + + + 引数 + + + z + + ベクトルまたは行列 + + + + y + + ベクトルまたは行列 (xと同じ大きさ) + + + + + + 説明 + + erfi は,-i \operatorname{erf}(ix) + で定義される xの虚数誤差関数を計算します. + + + + 例 + + + + アルゴリズム + + この関数はFaddeevaパッケージ ライブラリにもとづいています. + + + + 参照 + + + erf + + + erfcx + + + erfc + + + calerf + + + + + 履歴 + + + 5.5.0 + 関数erfiが複素数引数をサポート + + + + diff --git a/modules/special_functions/help/ja_JP/erfinv.xml b/modules/special_functions/help/ja_JP/erfinv.xml new file mode 100755 index 000000000..361de6acb --- /dev/null +++ b/modules/special_functions/help/ja_JP/erfinv.xml @@ -0,0 +1,88 @@ + + + + + erfinv + 逆誤差関数 + + + 呼び出し手順 + y = erfinv(x) + + + 引数 + + + x + + 実数ベクトルまたは行列 + + + + y + + (xと同じ大きさの)実数ベクトルまたは行列 + + + + + + 説明 + + erfinv 関数はerf + 誤差関数の逆, + つまり,-1 ≤ x ≤ 1 となるような + 任意のxについて + erf(erfinv(x)) = x を計算します. + + + + 例 + + + x = linspace(-0.99, 0.99, 100); + y = erfinv(x); + plot2d(x, y); + + + + 参照 + + + cdfnor + + + erf + + + erfc + + + erfcx + + + + + 参考文献 + + Milton Abramowitz and Irene A. Stegun, eds. Handbook of + Mathematical Functions with Formulas, Graphs, and Mathematical + Tables. New York: Dover, 1972. + + + diff --git a/modules/special_functions/help/ja_JP/findm.xml b/modules/special_functions/help/ja_JP/findm.xml new file mode 100755 index 000000000..a6a2539e0 --- /dev/null +++ b/modules/special_functions/help/ja_JP/findm.xml @@ -0,0 +1,27 @@ + + + + findm + 楕円フィルタ設計用 + + + 呼び出し手順 + [m]=findm(chi) + + + 説明 + + chi = %k(1-m)/%k(m)となるようなmを + 探索します + (find_freqで使用されます). + + + + 参照 + + + %k + + + + diff --git a/modules/special_functions/help/ja_JP/gamma.xml b/modules/special_functions/help/ja_JP/gamma.xml new file mode 100755 index 000000000..efd331ca4 --- /dev/null +++ b/modules/special_functions/help/ja_JP/gamma.xml @@ -0,0 +1,108 @@ + + + + +gamma +ガンマ関数. + + +呼び出し手順 +y = gamma(x) + + +引数 + + + x + + 実数ベクトルまたは行列 + + + + y + + (xと同じ大きさの)実数ベクトルまたは行列. + + + + + +説明 + + gamma(x) は, + xの全要素についてガンマ関数を計算します. + ガンマ関数は以下のように定義されます: + + + + + + + + +そして,階乗関数を実数に一般化します. + (gamma(n+1) = n!). + + + +例 + + + + a = -3; b = 5; + x = linspace(a,b,40000)'; + y = gamma(x); + plot2d(x, y, style=0, axesflag=5, rect=[a, -10, b, 10]) + xtitle("The gamma function on ["+string(a)+","+string(b)+"]") + + + +参照 + + + gammaln + + + dlgamma + + + + +履歴 + + + 5.4.0 + + list, mlist, tlistおよびハイパー行列型のオーバーロードが + 可能となりました. + + + + + diff --git a/modules/special_functions/help/ja_JP/gammaln.xml b/modules/special_functions/help/ja_JP/gammaln.xml new file mode 100755 index 000000000..d4e0f31ee --- /dev/null +++ b/modules/special_functions/help/ja_JP/gammaln.xml @@ -0,0 +1,77 @@ + + + + +gammaln +ガンマ関数の対数. + + +呼び出し手順 +y = gammaln(x) + + +引数 + + + x + + 実数ベクトル + + + + y + + 同じ大きさの実数ベクトル. + + + + + +説明 + + gammaln(x) はx + の全ての要素に関してオーバーフローおよびアンダーフローを回避しつつ + ガンマ関数の対数を計算します. + xは実数とする必要があります. + + + +例 + + + +参照 + + + gamma + + + dlgamma + + + + +履歴 + + + 5.4.0 + + list, mlist, tlistおよびハイパー行列型のオーバーロードが + 可能となりました. + + + + + diff --git a/modules/special_functions/help/ja_JP/legendre.xml b/modules/special_functions/help/ja_JP/legendre.xml new file mode 100755 index 000000000..70e82303a --- /dev/null +++ b/modules/special_functions/help/ja_JP/legendre.xml @@ -0,0 +1,181 @@ + + + + + legendre + 随伴ルジャンドル関数 + + + 呼び出し手順 + y = legendre(n,m,x [,normflag]) + + + パラメータ + + + n + + 非負の整数または等間隔で増分刻みが1の + 非負の整数のベクトル + + + + + m + + 非負の整数または等間隔で増分刻みが1の + 非負の整数のベクトル + + + + + x + + + 実数 (行) ベクトル (xの要素は + (-1,1)の範囲にある必要があります) + + + + + normflag + + (オプション) スカラー文字列 + + + + + + 説明 + + n および m がスカラーの場合, + legendre(n,m,x) は, + xの全要素について + 随伴ルジャンドル関数Pnm(x)を計算します. + 使用される定義を以下に示します: + + + + + + + + + + ただし,Pnはn次の + ルジャンドル多項式です. + legendre(n,0,x) は + xの全要素について + ルジャンドル関数Pn(x)を計算します. + + + normflagが"norm"に等しい時, + ((-1)^m係数を付けずに) + 正規化された出力が得られます : + + + + + + + + + これは,球面調和関数を計算する際に有用です(例3参照): + 効率化のため, + 最初の2つの引数の一つをベクトルとすることができ, + 例えば,legendre(n1:n2,0,x)は + xの要素における + n1, n1+1, ..., n2次の + ルジャンドル多項式を全て計算します. + また, + legendre(n,m1:m2,x) は + xにおいてm=m1, m1+1, ..., m2 + に関する随伴ルジャンドル関数Pnmを全て計算します. + + + + 出力形式 + + どの場合でも, yの形式は以下のようになります : + + + and : + + + xは行ベクトルの方が好ましいですが, + 任意のmx x nx行列を指定すると, + 1 x (mx * nx)行列とみなされ, + 以下のように列順に成形されます. + + + + 例 + = 0 then + y = (-1)^m/(sqrt(2*%pi))*exp(%i*m*phi)*legendre(l, m, cos(theta), "norm") + else + y = 1/(sqrt(2*%pi))*exp(%i*m*phi)*legendre(l, -m, cos(theta), "norm") + end +endfunction +// 3.2 : 他の有用な関数を定義 +function [x,y,z] = sph2cart(theta,phi,r) + // theta 行ベクトル 1 x nt + // phi 列ベクトル np x 1 + // r スカラーまたは np x nt 行列 (r(i,j) phi(i) theta(j)) における長さ + x = r.*(cos(phi)*sin(theta)); + y = r.*(sin(phi)*sin(theta)); + z = r.*(ones(phi)*cos(theta)); +endfunction +// 3-3 Y31(theta,phi)をプロット +l = 3; m = 1; +theta = linspace(0.1,%pi-0.1,60); +phi = linspace(0,2*%pi,120)'; +f = Y(l,m,theta,phi); +[x1,y1,z1] = sph2cart(theta,phi,abs(f)); [xf1,yf1,zf1] = nf3d(x1,y1,z1); +[x2,y2,z2] = sph2cart(theta,phi,abs(real(f))); [xf2,yf2,zf2] = nf3d(x2,y2,z2); +[x3,y3,z3] = sph2cart(theta,phi,abs(imag(f))); [xf3,yf3,zf3] = nf3d(x3,y3,z3); +clf() +subplot(1,3,1) +plot3d(xf1,yf1,zf1,flag=[2 4 4]); xtitle("|Y31(theta,phi)|") +subplot(1,3,2) +plot3d(xf2,yf2,zf2,flag=[2 4 4]); xtitle("|Real(Y31(theta,phi))|") +subplot(1,3,3) +plot3d(xf3,yf3,zf3,flag=[2 4 4]); xtitle("|Imag(Y31(theta,phi))|") + ]]> + + diff --git a/modules/special_functions/help/ja_JP/oldbessel.xml b/modules/special_functions/help/ja_JP/oldbessel.xml new file mode 100755 index 000000000..c0f309507 --- /dev/null +++ b/modules/special_functions/help/ja_JP/oldbessel.xml @@ -0,0 +1,186 @@ + + + + + oldbesseli + 第1種の修正ベッセル関数 (I_alpha). + + + oldbesselj + 第1種のベッセル関数 (J_alpha). + + + oldbesselk + 第2種の修正ベッセル関数 (K_alpha). + + + oldbessely + 第2種のベッセル関数 (Y_alpha). + + + 呼び出し手順 + y = oldbesseli(alpha,x) + y = oldbesseli(alpha,x,ice) + y = oldbesselj(alpha,x) + y = oldbesselk(alpha,x) + y = oldbesselk(alpha,x,ice) + y = oldbessely(alpha,x) + + + + パラメータ + + + x + + real vector with non negative entries + + + + alpha + + real vector with non negative entries regularly spaced with + increment equal to one + alpha=alpha0+(n1:n2) + + + + + ice + + integer flag, with default value 1 + + + + + + 説明 + これらの関数は古い関数であり, + besseli, besselj, + besselk, bessely を代わりに使用してください. + しかし,これらの2組の関数の構文は異なっていることに注意してください. + + + oldbesseli(alpha,x) computes modified Bessel + functions of the first kind (I sub alpha), for real, non-negative order + alpha and real non negative argument + x. besseli(alpha,x,2) computes + besseli(alpha,x).*exp(-x). + + + oldbesselj(alpha,x) computes Bessel functions of + the first kind (J sub alpha), for real, non-negative order + alpha and real non negative argument + x. + + + oldbesselk(alpha,x) computes modified Bessel + functions of the second kind (K sub alpha), for real, non-negative order + alpha and real non negative argument + x. besselk(alpha,x,2) computes + besselk(alpha,x).*exp(x). + + + oldbessely(alpha,x) computes Bessel functions of + the second kind (Y sub alpha), for real, non-negative order + alpha and real non negative argument + x. + + + alpha and x may be vectors. + The output is m-by-n with m + = size(x,'*') + + ,n = size(alpha,'*') whose + (i,j) entry is + oldbessel?(alpha(j),x(i)). + + + + Remarks + Y_alpha and J_alpha Bessel functions are 2 independant solutions of + the Bessel 's differential equation : + + + + + + + + + K_alpha and I_alpha modified Bessel functions are 2 independant + solutions of the modified Bessel 's differential equation : + + + + + + + + + + + Examples + 0+ +y = bessely(0:4,x); +clf() +plot2d(x,y, style=0:4, leg="Y0@Y1@Y2@Y3@Y4", rect=[0,-1.5,40,0.6]) +xtitle("Some Bessel functions of the second kind") + ]]> + + diff --git a/modules/special_functions/help/ja_JP/percentk.xml b/modules/special_functions/help/ja_JP/percentk.xml new file mode 100755 index 000000000..3e6513f3f --- /dev/null +++ b/modules/special_functions/help/ja_JP/percentk.xml @@ -0,0 +1,61 @@ + + + + %k + ヤコビの完全楕円積分 + + + 呼び出し手順 + [K]=%k(m) + + + パラメータ + + + m + + + 楕円積分のパラメータ 0<m<1 + (m はベクトルとすることができます) + + + + + K + + + 実軸上の 0 から 1までの楕円積分の値 + + + + + + + 説明 + + ヤコビの第一種完全楕円積分を計算します : + + + + 参考文献 + + Abramowitz and Stegun page 598 + + + + 例 + + + + 参照 + + + delip + + + + diff --git a/modules/special_functions/help/ja_JP/percentsn.xml b/modules/special_functions/help/ja_JP/percentsn.xml new file mode 100755 index 000000000..d5f454a1c --- /dev/null +++ b/modules/special_functions/help/ja_JP/percentsn.xml @@ -0,0 +1,80 @@ + + + + %sn + ヤコビ楕円関数 + + + 呼び出し手順 + [y]=%sn(x,m) + + + パラメータ + + + x + + + 楕円積分により定義される基本矩形の中の点; + x は複素数ベクトル + + + + + m + + + 楕円積分のパラメータ (0<m<1) + + + + + y + + 結果 + + + + + + 説明 + + パラメータmを指定した + ヤコビの楕円関数: + パラメータmの楕円積分の逆. + + + ゲインamはFortran形式で計算され, + 楕円関数用の加算式が適用されます. + + + + 例 + + + + 参照 + + + delip + + + %k + + + + diff --git a/modules/special_functions/help/mml/bessel_equation1.mml b/modules/special_functions/help/mml/bessel_equation1.mml new file mode 100755 index 000000000..39878a051 --- /dev/null +++ b/modules/special_functions/help/mml/bessel_equation1.mml @@ -0,0 +1,71 @@ + + + + + + + + + + + x + 2 + + ⋅ + + + + d + 2 + + y + + + dx + 2 + + + + + + + x + ⋅ + + dy + dx + + + + + + + + ( + + + x + 2 + + − + + α + 2 + + + ) + + ⋅ + y + + + = + 0, + + + α + ≥ + 0 + + + x^2 cdot {{d^2 y} over {dx^2}} + x cdot {{dy} over {dx}} + (x^2 - %alpha^2) cdot y = 0, %alpha>=0 + + \ No newline at end of file diff --git a/modules/special_functions/help/mml/bessel_equation2.mml b/modules/special_functions/help/mml/bessel_equation2.mml new file mode 100755 index 000000000..f62d24698 --- /dev/null +++ b/modules/special_functions/help/mml/bessel_equation2.mml @@ -0,0 +1,71 @@ + + + + + + + + + + + x + 2 + + ⋅ + + + + d + 2 + + y + + + dx + 2 + + + + + + + x + ⋅ + + dy + dx + + + + − + + + ( + + + x + 2 + + + + + α + 2 + + + ) + + ⋅ + y + + + = + 0, + + + α + ≥ + 0 + + + x^2 cdot {{d^2 y} over {dx^2}} + x cdot {{dy} over {dx}} - (x^2 + %alpha^2) cdot y = 0, %alpha >= 0 + + \ No newline at end of file diff --git a/modules/special_functions/help/mml/bessel_equation3.mml b/modules/special_functions/help/mml/bessel_equation3.mml new file mode 100755 index 000000000..de5200b16 --- /dev/null +++ b/modules/special_functions/help/mml/bessel_equation3.mml @@ -0,0 +1,98 @@ + + + + + + + + + + H + α + 1 + + + + ( + z + ) + + = + + J + α + + + + + ( + z + ) + + + + + i + ⋅ + + Y + α + + + + + ( + z + ) + + + + + + + H + α + 2 + + + + ( + z + ) + + = + + J + α + + + + + ( + z + ) + + − + + i + ⋅ + + Y + α + + + + + ( + z + ) + + + + + + alignl stack { +H^1_%alpha(z) = J_%alpha(z) + i cdot Y_%alpha(z) # +H^2_%alpha(z) = J_%alpha(z) - i cdot Y_%alpha(z) +} + + \ No newline at end of file diff --git a/modules/special_functions/help/mml/beta_equation1.mml b/modules/special_functions/help/mml/beta_equation1.mml new file mode 100755 index 000000000..531704422 --- /dev/null +++ b/modules/special_functions/help/mml/beta_equation1.mml @@ -0,0 +1,94 @@ + + + + + + B + + + + ( + + x + , + y + + ) + + = + + + + + ∫ + 0 + 1 + + + t + + x + − + 1 + + + + ⋅ + + + ( + + 1 + − + t + + ) + + + y + − + 1 + + + + ⋅ + dt + + + = + + + Γ + + + ( + x + ) + + ⋅ + Γ + + + ( + y + ) + + + + Γ + + ( + + x + + + y + + ) + + + + + + B(x,y) = int from 0 to 1 t^{x-1} cdot (1-t)^{y-1} cdot dt = {{%GAMMA(x) cdot %GAMMA(y)} over {%GAMMA(x + y)}} + + \ No newline at end of file diff --git a/modules/special_functions/help/mml/dlgamma_equation1.mml b/modules/special_functions/help/mml/dlgamma_equation1.mml new file mode 100755 index 000000000..5b70cb34c --- /dev/null +++ b/modules/special_functions/help/mml/dlgamma_equation1.mml @@ -0,0 +1,58 @@ + + + + + + + 1 + + Γ + + ( + x + ) + + + + + + + d + Γ + + ( + x + ) + + + dx + + = + + d + dx + + + + ( + + ln + + ( + + Γ + + ( + x + ) + + + ) + + + ) + + + {1 over {%GAMMA(x)}}{{d %GAMMA(x)} over {dx}} = {d over dx} (ln(%GAMMA(x))) + + \ No newline at end of file diff --git a/modules/special_functions/help/mml/gamma_equation1.mml b/modules/special_functions/help/mml/gamma_equation1.mml new file mode 100755 index 000000000..85df2557b --- /dev/null +++ b/modules/special_functions/help/mml/gamma_equation1.mml @@ -0,0 +1,50 @@ + + + + + + Γ + + + ( + x + ) + + = + + + + + ∫ + 0 + + + + ∞ + + + + t + + x + − + 1 + + + + ⋅ + + e + + − + t + + + + ⋅ + dt + + + + %GAMMA(x) = int from 0 to {+infty} t^{x-1} cdot e^{-t} cdot dt + + \ No newline at end of file diff --git a/modules/special_functions/help/mml/legendre_equation1.mml b/modules/special_functions/help/mml/legendre_equation1.mml new file mode 100755 index 000000000..50c88a10a --- /dev/null +++ b/modules/special_functions/help/mml/legendre_equation1.mml @@ -0,0 +1,77 @@ + + + + + + + P + n + m + + + + ( + x + ) + + = + + + + + ( + + − + 1 + + ) + + m + + ⋅ + + + ( + + 1 + − + + x + 2 + + + ) + + + m + / + 2 + + + + ⋅ + + + d + m + + + dx + m + + + + + + P + n + + + ( + x + ) + + + P_n^m(x) = (-1)^m cdot (1-x^2)^{m/2} cdot {d^m over dx^m}P_n(x) + + \ No newline at end of file diff --git a/modules/special_functions/help/mml/legendre_equation2.mml b/modules/special_functions/help/mml/legendre_equation2.mml new file mode 100755 index 000000000..8ce2da506 --- /dev/null +++ b/modules/special_functions/help/mml/legendre_equation2.mml @@ -0,0 +1,113 @@ + + + + + + + P + n + m + + + + ( + + x + , + norm + + ) + + = + + + + + ( + + + + 2n + + + 1 + + 2 + + ⋅ + + + + ( + + n + − + m + + ) + + ! + + + + ( + + n + + + m + + ) + + ! + + + + ) + + + ⋅ + + + ( + + 1 + − + + x + 2 + + + ) + + + m + / + 2 + + + + ⋅ + + + d + m + + + dx + m + + + + + + P + n + + + ( + x + ) + + + P_n^m(x,"norm") = {sqrt({{2n+1} over {2}} cdot {{(n-m)!} over {(n+m)!}})} cdot (1-x^2)^{m/2} cdot {d^m over dx^m}P_n(x) + + \ No newline at end of file diff --git a/modules/special_functions/help/pt_BR/addchapter.sce b/modules/special_functions/help/pt_BR/addchapter.sce new file mode 100755 index 000000000..891fde6e7 --- /dev/null +++ b/modules/special_functions/help/pt_BR/addchapter.sce @@ -0,0 +1,11 @@ +// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +// Copyright (C) 2009 - DIGITEO +// +// This file must be used under the terms of the CeCILL. +// This source file is licensed as described in the file COPYING, which +// you should have received as part of this distribution. The terms +// are also available at +// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt + +add_help_chapter("Funções Especiais",SCI+"/modules/special_functions/help/pt_BR",%T); + diff --git a/modules/special_functions/help/pt_BR/amell.xml b/modules/special_functions/help/pt_BR/amell.xml new file mode 100755 index 000000000..0a63ccc9e --- /dev/null +++ b/modules/special_functions/help/pt_BR/amell.xml @@ -0,0 +1,66 @@ + + + + + amell + funo "am" de Jacobi + + + Seqncia de Chamamento + [sn]=amell(u,k) + + + Parmetros + + + u + + escalar real ou vetor de reais + + + + k + + escalar + + + + sn + + escalar real ou vetor de reais + + + + + + Descrio + + Computa a funo elptica de Jacobi am(u,k) onde + k o parmetro e u o argumento. + Se u um vetor sn o vetor dos + valores computados (elemento a elemento) . Usado na funo + %sn. + + + + Ver Tambm + + + delip + + + %sn + + + + diff --git a/modules/special_functions/help/pt_BR/bessel.xml b/modules/special_functions/help/pt_BR/bessel.xml new file mode 100755 index 000000000..ce6abc8c9 --- /dev/null +++ b/modules/special_functions/help/pt_BR/bessel.xml @@ -0,0 +1,256 @@ + + + + + besseli + funes modificadas de Bessel do primeiro tipo (I sub + alfa). + + + + besselj + funes de Bessel do primeiro tipo (J sub alpha). + + + besselk + funes modificadas de Bessel do segundo tipo (K sub + alpha). + + + + bessely + funes de Bessel do segundo tipo (Y sub alpha). + + + besselh + funes de Bessel do terceiro tipo (tambm conhecidas como + funes de Hankel) + + + + Seqncia de Chamamento + y = besseli(alpha,x [,ice]) + y = besselj(alpha,x [,ice]) + y = besselk(alpha,x [,ice]) + y = bessely(alpha,x [,ice]) + y = besselh(alpha,x) + y = besselh(alpha,K,x [,ice]) + + + + Parmetros + + + x + + vetor de reais ou complexos + + + + alpha + + vetor de reais + + + + ice + + flag (sinalizador) inteiro, com valor padro 0 + + + + K + + inteiro, com valores possveis 1 ou 2, a funo do tipo de + Hankel. + + + + + + + Descrio + + + + besseli(alpha,x) computa as funes de Bessel + modificadas do primeiro tipo (I sub alfa), para ordem real + alpha e argumento x. + besseli(alpha,x,1) computa + besseli(alpha,x).*exp(-abs(real(x))). + + + + + besselj(alpha,x) computa as funes de Bessel + do primeiro tipo (J sub alfa), para ordem real + alpha e argumento x. + besselj(alpha,x,1) computa + besselj(alpha,x).*exp(-abs(imag(x))). + + + + + besselk(alpha,x) computa as funes de Bessel + modificadas do segundo tipo (K sub alfa), para ordem real + alpha e argumento x. + besselk(alpha,x,1) computa + besselk(alpha,x).*exp(x). + + + + + bessely(alpha,x) computa as funes de Bessel + do segundo tipo (Y sub alfa), para ordem real alpha + e argumento x. + bessely(alpha,x,1) computa + bessely(alpha,x).*exp(-abs(imag(x))). + + + + + besselh(alpha [,K] ,x) computa as funes de + Bessel do terceiro tipo (funo de Hankel H1 ou H2, dependendo do + K), para ordem real alpha e + argumentot x. Se omitido, K + suposto como sendo 1. besselh(alpha,1,x,1) computa + besselh(alpha,1,x).*exp(-%i*x) e + besselh(alpha,2,x,1) computa + besselh(alpha,2,x).*exp(%i*x) + + + + + + Observaes + + Se alpha e x so arrays de + mesmo tamanho, o resultado y tambm ter este tamanho. + Se uma entrada um escalar, ela expandida para o tamanho da outra + entrada. Se uma entrada um vetor linha e a outra um vetor coluna, o + resultado y um table 2-dimensional ("tabela") de + valores de funes. + + As funes de Bessel Y_alfa e J_alfa so duas solues independentes + da equao diferencial de Bessel: + + = 0 + ]]> + As funes modificadas de Bessel K_alfa e I_alfa so duas solues + independentes para a equao diferencial de Bessel : + + = 0 + ]]> + As funes de Hankel de primeiro e segundo tipos H^1_alfa e + H^2_alfa, so combinaes lineares das funes de Bessel de primeiro e + segundo tipos: + + + + + Exemplos + 0 & y2 ~= 0); +clf() +subplot(2,1,1) +plot2d(x,y1,style=2) +xtitle("besselj(0.5,x)") +subplot(2,1,2) +plot2d(x(ind), er(ind), style=2, logflag="nl") +xtitle("Erro relativo entre as duas frmulas para besselj(0.5,x)") + +// Funes K de Bessel +// ================= +x = linspace(0.01,10,5000)'; +clf() +subplot(2,1,1) +plot2d(x,besselk(0:4,x), style=0:4, rect=[0,0,6,10]) +legend('K'+string(0:4),1); +xtitle("Algumas funes modificadas de Bessel do segundo tipo") +subplot(2,1,2) +plot2d(x,besselk(0:4,x,1), style=0:4, rect=[0,0,6,10]) +legend('K'+string(0:4),1); +xtitle("Algumas funes modificadas de Bessel do segundo tipo escaladas") + +// Funes Y de Bessel +// ================= +x = linspace(0.1,40,5000)'; // funes Y de Bessel no possuem limite para x -> 0+ +clf() +plot2d(x,bessely(0:4,x), style=0:4, rect=[0,-1.5,40,0.6]) +legend('Y'+string(0:4),4); +xtitle("Algumas funes de Bessel do segundo tipo") + +// Funes H de Bessel +// ================= +x=-4:0.025:2; y=-1.5:0.025:1.5; +[X,Y] = ndgrid(x,y); +H = besselh(0,1,X+%i*Y); +clf();f=gcf(); +xset("fpf"," ") +f.color_map=jetcolormap(16); +contour2d(x,y,abs(H),0.2:0.2:3.2,strf="034",rect=[-4,-1.5,3,1.5]) +legends(string(0.2:0.2:3.2),1:16,"ur") +xtitle("Curvas de nvel de |H1(0,z)|") + ]]> + + + Autores + + Amos, D. E., (SNLA) + Daniel, S. L., (SNLA) + Weston, M. K., (SNLA) + + + + Funo Usada + Os cdigos-fontes podem ser achados em SCI/modules/special_functions/src/fortran/slatec e SCI/modules/special_functions/src/fortran + Slatec : dbesi.f, zbesi.f, dbesj.f, zbesj.f, dbesk.f, zbesk.f, + dbesy.f, zbesy.f, zbesh.f + + Drivers para estender a rea de definio (Serge Steer INRIA): + dbesig.f, zbesig.f, dbesjg.f, zbesjg.f, dbeskg.f, zbeskg.f, dbesyg.f, + zbesyg.f, zbeshg.f + + + diff --git a/modules/special_functions/help/pt_BR/beta.xml b/modules/special_functions/help/pt_BR/beta.xml new file mode 100755 index 000000000..51be052cd --- /dev/null +++ b/modules/special_functions/help/pt_BR/beta.xml @@ -0,0 +1,106 @@ + + + + + beta + funo beta + + + Seqncia de Chamamento + z = beta(x,y) + + + Parmetros + + + x, y + + dois reais positivos ou duas matrizes (ou vetores) de reais + positivos de mesmo tamanho + + + + + z + + um real ou uma matriz de reais com mesmo tamanho que + x com z(i,j) = + beta(x(i,j),y(i,j)) + + . + + + + + + + Descrio + Computa a funo beta completa : + + + Para x e y pequenos, o + algoritmo usa a expresso em funo da funo gama, de outro modo, ele + aplica a funo exponencial no resutado da funo + betaln function fornecido no DCDFLIB: Biblioteca de + Rotinas FORTRAN para Funes, Inversas e Outros Parmetros de Distribuio + Cumulativa (ver cdfbet para maiores + informaes sobre DCDFLIB). + + + + Exemplos + + + + Ver Tambm + + + gamma + + + cdfbet + + + + diff --git a/modules/special_functions/help/pt_BR/calerf.xml b/modules/special_functions/help/pt_BR/calerf.xml new file mode 100755 index 000000000..2d774eb79 --- /dev/null +++ b/modules/special_functions/help/pt_BR/calerf.xml @@ -0,0 +1,78 @@ + + + + + calerf + computa funes de erro + + + Parmetros + + + x + + matriz ou vetor de reais + + + + flag + + indicador inteiro + + + + y + + matriz ou vetor de reais (de mesmo tamanho que x) + + + + + + Descrio + + calerf(x,0) computa a funo de erro + erf(x) + + + calerf(x,1) computa a funo de erro complementar + erfc(x) + + + calerf(x,2) computa a funo de erro complementar + escalada erfcx(x) + + + + Exemplos + + + + Ver Tambm + + + erf + + + erfc + + + erfcx + + + + diff --git a/modules/special_functions/help/pt_BR/delip.xml b/modules/special_functions/help/pt_BR/delip.xml new file mode 100755 index 000000000..f32677a42 --- /dev/null +++ b/modules/special_functions/help/pt_BR/delip.xml @@ -0,0 +1,143 @@ + + + + + delip + Integral elíptica + + + Seqüência de Chamamento + [r]=delip(x,ck) + + + Parâmetros + + + x + + vetor real com elementos não-negativos + + + + ck + + número real entre -1 e 1 + + + + r + + número real ou complexo ou vetor de reais ou complexos com + mesmo tamanho que x + + + + + + + Descrição + A integral elíptica de primeira espécie com parâmetro + ck definido como segue: + + + + + + + + + + ∫ + 0 + x + + + dt + + + + ( + + 1 + − + + t + 2 + + + ) + + + ( + + + 1 + − + + c + k + 2 + + + + t + 2 + + + ) + + + + + + int from{0} to{x} {{dt} over sqrt{ + (1-t^2)(1-c_k^2 t^2)}} + + + + + + + + + Onde x é real e positivo e ck + está em [-1 1]. + + Se x é menor do que ou igual a 1, o resultado é real. + + Quando chamado com x, um vetor real + r é avaliado para cada entrada de + x. + + + + Exemplos + + + + Ver Também + + + amell + + + %sn + + + + diff --git a/modules/special_functions/help/pt_BR/dlgamma.xml b/modules/special_functions/help/pt_BR/dlgamma.xml new file mode 100755 index 000000000..af823e413 --- /dev/null +++ b/modules/special_functions/help/pt_BR/dlgamma.xml @@ -0,0 +1,71 @@ + + + + + dlgamma + derivada da funo gammaln, funo psi + + + Seqncia de Chamamento + y = dlgamma(x) + + + Parmetros + + + x + + vetor de reais + + + + y + + vetor de reais com o mesmo tamanho + + + + + + Descrio + + dlgamma(x) avalia, em todos os elementos de + x a derivada logartmica da funo gama (gamma), que + corresponde tambm derivada da funo ln(gama) (gammaln): + + + + x deve ser real. Tambm conhecida como a funo + psi. + + + + Exemplos + + + + Ver Tambm + + + gamma + + + gammaln + + + + diff --git a/modules/special_functions/help/pt_BR/erf.xml b/modules/special_functions/help/pt_BR/erf.xml new file mode 100755 index 000000000..9cdca9cc1 --- /dev/null +++ b/modules/special_functions/help/pt_BR/erf.xml @@ -0,0 +1,78 @@ + + + + + erf + funo de erro + + + Seqncia de Chamamento + y = erf(x) + + + Parmetros + + + x + + vetor ou matriz de reais + + + + y + + vetor ou matriz de reais (de mesmo tamanho que x) + + + + + + Descrio + + erf computa a funo de erro: + + + + + Exemplos + + + + Ver Tambm + + + calerf + + + cdfnor + + + erfc + + + erfcx + + + erfinv + + + + diff --git a/modules/special_functions/help/pt_BR/erfc.xml b/modules/special_functions/help/pt_BR/erfc.xml new file mode 100755 index 000000000..551d01ad0 --- /dev/null +++ b/modules/special_functions/help/pt_BR/erfc.xml @@ -0,0 +1,75 @@ + + + + + erfc + funo de erro complementar + + + Seqncia de Chamamento + y = erfc(x) + + + Parmetros + + + x + + real vector or matrix + + + + y + + real vector or matrix (of same size than x) + + + + + + Descrio + + erfc computa a funo de erro + complementar: + + + + + Exemplos + + + + Ver Tambm + + + erf + + + erfcx + + + calerf + + + + diff --git a/modules/special_functions/help/pt_BR/erfcx.xml b/modules/special_functions/help/pt_BR/erfcx.xml new file mode 100755 index 000000000..3337e7257 --- /dev/null +++ b/modules/special_functions/help/pt_BR/erfcx.xml @@ -0,0 +1,68 @@ + + + + + erfcx + funo de erro complementar escalada + + + Seqncia de Chamamento + y = erfcx(x) + + + Parameters + + + x + + vetor ou matriz de reais + + + + y + + vetor ou matriz de reais (de mesmo tamanho que x) + + + + + + Descrio + + erfcx computa a funo de erro complementar + escalada: + + --------- quando x --> +oo + x sqrt(pi) + ]]> + + + Ver Tambm + + + erf + + + erfc + + + calerf + + + + diff --git a/modules/special_functions/help/pt_BR/erfinv.xml b/modules/special_functions/help/pt_BR/erfinv.xml new file mode 100755 index 000000000..ea8bdb039 --- /dev/null +++ b/modules/special_functions/help/pt_BR/erfinv.xml @@ -0,0 +1,83 @@ + + + + + erfinv + função inversa à função de erro + + + Seqüência de Chamamento + y = erfinv(x) + + + Parâmetros + + + x + + vetor ou matriz de reais + + + + y + + vetor ou matriz de reais (de mesmo tamanho que x) + + + + + + Descrição + + erfinv computa a função inversa à função de + erro erf. x = erfinv(y) + satisfaz y = erf(x), -1 ≤ y ≤ + 1 + + ,∞ ≤ x ≤ ∞. + + + + Exemplos + + + + Ver Também + + + cdfnor + + + erf + + + erfc + + + erfcx + + + + + Referências + Milton Abramowitz e Irene A. Stegun, eds. Handbook of Mathematical + Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, + 1972. + + + diff --git a/modules/special_functions/help/pt_BR/gamma.xml b/modules/special_functions/help/pt_BR/gamma.xml new file mode 100755 index 000000000..75cee9b3b --- /dev/null +++ b/modules/special_functions/help/pt_BR/gamma.xml @@ -0,0 +1,92 @@ + + + + + gamma + funo gama + + + Seqncia de Chamamento + y = gamma(x) + + + Parmetros + + + x + + vetor ou matriz de reais ou complexos + + + + y + + vetor ou matriz de reais ou complexos de mesmo tamanho que + x + + + + + + + Descrio + + gamma(x) avalia a funo gama em todos os + elementos de x. A funo gama defininda por : + + + e generaliza a funo fatorial para os nmeros reais + (gamma(n+1) = n!). + + + + Exemplos + + + + Ver Tambm + + + gammaln + + + dlgamma + + + + + Autores + W. J. Cody e L. Stoltz (cdigo de Netlib (specfun)) + + diff --git a/modules/special_functions/help/pt_BR/gammaln.xml b/modules/special_functions/help/pt_BR/gammaln.xml new file mode 100755 index 000000000..7dfdb4ece --- /dev/null +++ b/modules/special_functions/help/pt_BR/gammaln.xml @@ -0,0 +1,68 @@ + + + + + gammaln + o logaritmo (natural) da funo gama + + + Seqncia de Chamamento + y = gammaln(x) + + + Parameters + + + x + + vetor de reais + + + + y + + vetor de reais com o mesmo tamanho + + + + + + Description + + gammaln(x) avalia o logaritmo (natural) da funo + gama em todos os elementos de x, evitando underflow e + overflow. x deve ser de reais. + + + + Exemplos + + + + Ver Tambm + + + gamma + + + dlgamma + + + + + Autores + W. J. Cody e L. Stoltz (cdigo de Netlib (specfun)) + + diff --git a/modules/special_functions/help/pt_BR/legendre.xml b/modules/special_functions/help/pt_BR/legendre.xml new file mode 100755 index 000000000..bd2a8f0e6 --- /dev/null +++ b/modules/special_functions/help/pt_BR/legendre.xml @@ -0,0 +1,189 @@ + + + + + legendre + funes associadas de Legendre + + + Seqncia de Chamamento + y = legendre(n,m,x [,normflag]) + + + Parmetros + + + n + + inteiro no-negativo ou vetor de inteiros no-negativos + igualmente espaados com incremento igual a 1 + + + + + m + + inteiro no-negativo ou vetor de inteiros no-negativos + igualmente espaados com incremento igual a 1 + + + + + x + + + vetor (linha) de reais (os elementos de x + devem estar no intervalo (-1,1) ) + + + + + normflag + + (opcional) escalar string + + + + + + Descrio + + Quando n e m so escalares, + legendre(n,m,x) avalia a funo de Legendre associada + Pnm(x) em todos os elementos de x. A definio usada + : + + + + onde Pn o polinmio de Legendre de grau + n. Ento, legendre(n,0,x) avalia o + polinmio de Legendre Pn(x) em todos os elementos de + x. + + Quando normflag igual a "norm" o resultado uma verso + normalizada (sem o fator (-1)^m ), precisamente : + + + que til para computar funes harmnicas esfricas (ver exemplo + 3): + + Por eficincia, um dos primeiros dois argumentos pode ser um vetor, + por exemplo legendre(n1:n2,0,x) avalia todos os + polinmios de Legendre de graus n1, n1+1, ..., n2 nos + elementos de x e legendre(n,m1:m2,x) + avalia todas as funes de Legendre associadas Pnm para m=m1, + m1+1, ..., m2 + + em x. + + + + Formato de Sada + + Em qualquer caso, o formato y : + + + e : + + + de tal modo que x preferivelmente um vetor + linha, mas qualquer matriz mx x nx excetuada e + considerada como uma matriz 1 x (mx * nx) matrix, + reformada segundo a ordem das colunas. + + + + Exemplos + = 0 then + y = (-1)^m/(sqrt(2*%pi))*exp(%i*m*phi)*legendre(l, m, cos(theta), "norm") + else + y = 1/(sqrt(2*%pi))*exp(%i*m*phi)*legendre(l, -m, cos(theta), "norm") + end +endfunction + +// 3.2 : definindo outra funo til +function [x,y,z] = sph2cart(theta,phi,r) + // vetor linha teta 1 x nt + // vetor coluna phi np x 1 + // r escalar ou matriz np x nt (r(i,j) o comprimento em phi(i) theta(j)) + x = r.*(cos(phi)*sin(theta)); + y = r.*(sin(phi)*sin(theta)); + z = r.*(ones(phi)*cos(theta)); +endfunction + +// 3-3 plot de Y31(theta,phi) +l = 3; m = 1; +theta = linspace(0.1,%pi-0.1,60); +phi = linspace(0,2*%pi,120)'; +f = Y(l,m,theta,phi); +[x1,y1,z1] = sph2cart(theta,phi,abs(f)); [xf1,yf1,zf1] = nf3d(x1,y1,z1); +[x2,y2,z2] = sph2cart(theta,phi,abs(real(f))); [xf2,yf2,zf2] = nf3d(x2,y2,z2); +[x3,y3,z3] = sph2cart(theta,phi,abs(imag(f))); [xf3,yf3,zf3] = nf3d(x3,y3,z3); + +clf() +subplot(1,3,1) +plot3d(xf1,yf1,zf1,flag=[2 4 4]); xtitle("|Y31(theta,phi)|") +subplot(1,3,2) +plot3d(xf2,yf2,zf2,flag=[2 4 4]); xtitle("|Real(Y31(theta,phi))|") +subplot(1,3,3) +plot3d(xf3,yf3,zf3,flag=[2 4 4]); xtitle("|Imag(Y31(theta,phi))|") + ]]> + + + Autores + + Smith, John M. (cdigo dxlegf.f de Slatec) + B. Pincon (interface Scilab) + + + diff --git a/modules/special_functions/help/ru_RU/addchapter.sce b/modules/special_functions/help/ru_RU/addchapter.sce new file mode 100755 index 000000000..8fbedc933 --- /dev/null +++ b/modules/special_functions/help/ru_RU/addchapter.sce @@ -0,0 +1,11 @@ +// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +// Copyright (C) 2009 - DIGITEO +// +// This file must be used under the terms of the CeCILL. +// This source file is licensed as described in the file COPYING, which +// you should have received as part of this distribution. The terms +// are also available at +// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt + +add_help_chapter("Special Functions",SCI+"/modules/special_functions/help/ru_RU",%T); + diff --git a/modules/special_functions/help/ru_RU/amell.xml b/modules/special_functions/help/ru_RU/amell.xml new file mode 100755 index 000000000..b63795fb4 --- /dev/null +++ b/modules/special_functions/help/ru_RU/amell.xml @@ -0,0 +1,65 @@ + + + + + amell + Эллиптическая функция am Якоби + + + Последовательность вызова + [sn]=amell(u,k) + + + Аргументы + + + u + + вещественный скаляр или вектор + + + + k + + скаляр + + + + sn + + вещественный скаляр или вектор + + + + + + Описание + + Вычисляет эллиптическую функцию am(u,k) Якоби, где + k - параметр, а u - аргумент. Если + u является вектором, то sn является вектором + вычисленных (поэлементно) значений. Используется в функции %sn. + + + + Смотрите также + + + delip + + + %sn + + + + diff --git a/modules/special_functions/help/ru_RU/delip.xml b/modules/special_functions/help/ru_RU/delip.xml new file mode 100755 index 000000000..7aecaf170 --- /dev/null +++ b/modules/special_functions/help/ru_RU/delip.xml @@ -0,0 +1,146 @@ + + + + + delip + полный и неполный эллиптический интеграл первого рода + + + Последовательность вызова + [r]=delip(x,ck) + + + Аргументы + + + x + + вещественный вектор/матрица с неотрицательными элементами real vector with non negative elements + + + + ck + + вещественное число между -1 и 1 + + + + r + + + вещественное или комплексное число (или вектор/матрица) того же + размера, что и x + + + + + + + Описание + + Эллиптический интеграл первого рода с параметром ck + определяется как: + + + + + + + + + + ∫ + 0 + x + + + dt + + + + ( + + 1 + − + + t + 2 + + + ) + + + ( + + + 1 + − + + c + k + 2 + + + + t + 2 + + + ) + + + + + + int from{0} to{x} {{dt} over sqrt{ + (1-t^2)(1-c_k^2 t^2)}} + + + + + + + + + Где x - вещественное положительное число, + ck - лежит на интервале [-1 1]. + + + Если x меньше 1, то результат вещественный. + + + Вектор/матрица r + вычисляется для каждого элемента x. + + + + Примеры + + + + Смотрите также + + + amell + + + %sn + + + + -- cgit