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## maxima-commits

 [Maxima-commits] CVS: maxima/doc/info Special.texi,1.4,1.5 From: Raymond Toy - 2002-06-25 17:12:57 Update of /cvsroot/maxima/maxima/doc/info In directory usw-pr-cvs1:/tmp/cvs-serv10078/doc/info Modified Files: Special.texi Log Message: Added rudimentary documentation for the functions BESSEL_J, BESSEL_Y, BESSEL_I, and BESSEL_K, and for the variable BESSELEXPAND. Index: Special.texi =================================================================== RCS file: /cvsroot/maxima/maxima/doc/info/Special.texi,v retrieving revision 1.4 retrieving revision 1.5 diff -u -d -r1.4 -r1.5 --- Special.texi 9 May 2002 14:12:04 -0000 1.4 +++ Special.texi 25 Jun 2002 17:12:54 -0000 1.5 @@ -101,6 +101,127 @@ J[I+A- ENTIER(A)](Z). @end defun + +@c @node BESSEL_J +@c @unnumberedsec phony +@defun BESSEL_J [v](z) +The Bessel function of the first kind of order @math{v} and argument +@math{z}. It is defined by +@ifinfo +@example + INF + ==== k - v - 2 k v + 2 k + \ (- 1) 2 z + > -------------------------- + / k! GAMMA(v + k + 1) + ==== + k = 0 +@end example +@end ifinfo + +@tex +$$\sum_{k=0}^{\infty }{{{\left(-1\right)^{k}\,\left(z\over 2\right)^{v+2\,k} + }\over{k!\,\Gamma\left(v+k+1\right)}}}$$ +@end tex +@end defun + +@c @node BESSEL_Y +@c @unnumberedsec phony +@defun BESSEL_Y [v](z) +The Bessel function of the second kind of order @math{v} and argument +@math{z}. It is defined by +@ifinfo +@example + COS(%PI v) BESSEL_J (z) - BESSEL_J (z) + v - v + ---------------------------------------- + SIN(%PI v) +@end example +@end ifinfo + +@tex +$${{\cos \left(\pi\,v\right)\,J_{v}(z)-J_{-v}(z)}\over{ + \sin \left(\pi\,v\right)}}$$ +@end tex + +when @math{v} is not an integer. When @math{v} is an integer @math{n}, +the limit as @math{v} approaches @math{n} is taken. + +@end defun + +@c @node BESSEL_I +@c @unnumberedsec phony +@defun BESSEL_I [v](z) +The modified Bessel function of the first kind of order @math{v} and +argument @math{z}. It is defined by +@ifinfo +@example + + INF + ==== - v - 2 k v + 2 k + \ 2 z +(D1) > ------------------- + / k! GAMMA(v + k + 1) + ==== + k = 0 +@end example +@end ifinfo + +@tex +$$\sum_{k=0}^{\infty } {{1\over{k!\,\Gamma + \left(v+k+1\right)}} {\left(z\over 2\right)^{v+2\,k}}}$$ +@end tex + +@end defun + +@c @node BESSEL_K +@c @unnumberedsec phony +@defun BESSEL_K [v](z) +The modified Bessel function of the second kind of order @math{v} and +argument @math{z}. It is defined by +@ifinfo +@example + + %PI CSC(%PI v) (BESSEL_I (z) - BESSEL_I (z)) + - v v +(D3) ---------------------------------------------- + 2 +@end example +@end ifinfo +@tex +$${{\pi\,\csc \left(\pi\,v\right)\,\left(I_{-v}(z)-I_{v}(z)\right)}\over{2}}$$ +@end tex +when @math{v} is not an integer. If @math{v} is an integer @math{n}, +then the limit as @math{v} approaches @math{n} is taken. +@end defun + +@defvar BESSELEXPAND + default: FALSE + Controls expansion of the Bessel functions when the order half of an + odd integer. In this case, the Bessel functions can be expanded in + terms of other elementary functions. When BESSELEXPAND is true, the + Bessel function is expanded. + +@example +(C1) bessel_j[3/2](z); + +(D1) BESSEL_J (z) + 3/2 +(C2) besselexpand:true; + +(D2) TRUE +(C3) bessel_j[3/2](z); + + COS(z) SIN(z) + SQRT(2) SQRT(z) (------ - ------) + z 2 + z +(D3) - --------------------------------- + SQRT(%PI) + +@end example +@end defvar + @c @node BETA @c @unnumberedsec phony @defun BETA (X, Y)