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<H1><A NAME="SEC48" HREF="maxima_toc.html#TOC48">Special Functions</A></H1>
<H2><A NAME="SEC49" HREF="maxima_toc.html#TOC49">Introduction to Special Functions</A></H2>
<P>
<DL>
<DT><U>%J:</U> <B></B>
<DD><A NAME="IDX419"></A>
[index](expr) - Bessel Funct 1st Kind (in SPECINT)
</P>
</DL>
<P>
<DL>
<DT><U>%K:</U> <B>[index](expr)</B>
<DD><A NAME="IDX420"></A>
Bessel Funct 2nd Kind (in SPECINT)
</P>
<P>
Constant, in ODE2
</P>
</DL>
<H2><A NAME="SEC50" HREF="maxima_toc.html#TOC50">GAMALG</A></H2>
<P>
- A Dirac gamma matrix algebra program which takes traces of
and does manipulations on gamma matrices in n dimensions. It may be
loaded into MACSYMA by
LOADFILE("gam");
A preliminary manual is contained in the file SHARE;GAM USAGE and may
be printed using
PRINTFILE(GAM,USAGE,SHARE);
</P>
<H2><A NAME="SEC51" HREF="maxima_toc.html#TOC51">SPECINT</A></H2>
<P>
- The Hypergeometric Special Functions Package HYPGEO is
still under development. At the moment it will find the Laplace
Transform or rather, the integral from 0 to INF of some special
functions or combinations of them. The factor, EXP(-P*var) must be
explicitly stated. The syntax is as follows:
SPECINT(EXP(-P*var)*expr,var);
where var is the variable of integration and expr may be
any expression containing special functions (at your own risk).
Special function notation follows:
<PRE>
%J[index](expr) Bessel Funct 1st Kind
%K[index](expr) " " 2nd Kind
%I[ ]( ) Modified Bessel
%HE[ ]( ) Hermite Poly
%P[ ]( ) Legendre Funct
%Q[ ]( ) Legendre of second kind
HSTRUVE[ ]( ) Struve H Function
LSTRUVE[ ]( ) " L Function
%F[ ]([],[],expr) Hypergeometric Function
GAMMA()
GAMMAGREEK()
GAMMAINCOMPLETE()
SLOMMEL
%M[]() Whittaker Funct 1st Kind
%W[]() " " 2nd "
</PRE>
<P>
For a better feeling for what it can do, do DEMO(HYPGEO,DEMO,SHARE1); .
</P>
<H2><A NAME="SEC52" HREF="maxima_toc.html#TOC52">Definitions for Special Functions</A></H2>
<P>
<DL>
<DT><U>Function:</U> <B>AIRY</B> <I>(X)</I>
<DD><A NAME="IDX421"></A>
returns the Airy function Ai of real argument X.
The file SHARE1;AIRY FASL contains routines to evaluate the Airy functions
Ai(X), Bi(X), and their derivatives dAi(X), dBi(X). Ai and Bi satisfy the
AIRY eqn diff(y(x),x,2)-x*y(x)=0. Read SHARE1;AIRY USAGE for details.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>ASYMP</B>
<DD><A NAME="IDX422"></A>
- A preliminary version of a program to find the asymptotic behavior
of Feynman diagrams has been installed on the SHARE1; directory.
For further information, see the file SHARE1;ASYMP USAGE.
(For Asymptotic Analysis functions, see ASYMPA.)
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>ASYMPA</B>
<DD><A NAME="IDX423"></A>
- Asymptotic Analysis - The file SHARE1;ASYMPA > contains
simplification functions for asymptotic analysis, including the big-O
and little-o functions that are widely used in complexity analysis and
numerical analysis. Do BATCH("asympa.mc"); .
(For asymptotic behavior of Feynman diagrams, see ASYMP.)
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>BESSEL</B> <I>(Z,A)</I>
<DD><A NAME="IDX424"></A>
returns the Bessel function J for complex Z and real A >
0.0 . Also an array BESSELARRAY is set up such that BESSELARRAY[I] =
J[I+A- ENTIER(A)](Z).
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>BETA</B> <I>(X, Y)</I>
<DD><A NAME="IDX425"></A>
same as GAMMA(X)*GAMMA(Y)/GAMMA(X+Y).
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>GAMMA</B> <I>(X)</I>
<DD><A NAME="IDX426"></A>
the gamma function. GAMMA(I)=(I-1)! for I a positive
integer. For the Euler-Mascheroni constant, see %GAMMA. See also the
MAKEGAMMA function. The variable GAMMALIM[1000000] (which see)
controls simplification of the gamma function.
</P>
</DL>
<P>
<DL>
<DT><U>Variable:</U> <B>GAMMALIM</B>
<DD><A NAME="IDX427"></A>
default: [1000000] controls simplification of the gamma
function for integral and rational number arguments. If the absolute
value of the argument is not greater than GAMMALIM, then
simplification will occur. Note that the FACTLIM switch controls
simplification of the result of GAMMA of an integer argument as well.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>INTOPOIS</B> <I>(A)</I>
<DD><A NAME="IDX428"></A>
converts A into a Poisson encoding.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>MAKEFACT</B> <I>(exp)</I>
<DD><A NAME="IDX429"></A>
transforms occurrences of binomial,gamma, and beta
functions in exp to factorials.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>MAKEGAMMA</B> <I>(exp)</I>
<DD><A NAME="IDX430"></A>
transforms occurrences of binomial,factorial, and beta
functions in exp to gamma functions.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>NUMFACTOR</B> <I>(exp)</I>
<DD><A NAME="IDX431"></A>
gives the numerical factor multiplying the expression
exp which should be a single term. If the gcd of all the terms in a
sum is desired the CONTENT function may be used.
<PRE>
(C1) GAMMA(7/2);
(D1) 15 SQRT(%PI)
------------
8
(C2) NUMFACTOR(%);
15
(D2) --
8
</PRE>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>OUTOFPOIS</B> <I>(A)</I>
<DD><A NAME="IDX432"></A>
converts A from Poisson encoding to general
representation. If A is not in Poisson form, it will make the
conversion, i.e. it will look like the result of
OUTOFPOIS(INTOPOIS(A)). This function is thus a canonical simplifier
for sums of powers of SIN's and COS's of a particular type.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>POISDIFF</B> <I>(A, B)</I>
<DD><A NAME="IDX433"></A>
differentiates A with respect to B. B must occur only
in the trig arguments or only in the coefficients.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>POISEXPT</B> <I>(A, B)</I>
<DD><A NAME="IDX434"></A>
B a positive integer) is functionally identical to
INTOPOIS(A**B).
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>POISINT</B> <I>(A, B)</I>
<DD><A NAME="IDX435"></A>
integrates in a similarly restricted sense (to
POISDIFF). Non-periodic terms in B are dropped if B is in the trig
arguments.
</P>
</DL>
<P>
<DL>
<DT><U>Variable:</U> <B>POISLIM</B>
<DD><A NAME="IDX436"></A>
default: [5] - determines the domain of the coefficients in
the arguments of the trig functions. The initial value of 5
corresponds to the interval [-2^(5-1)+1,2^(5-1)], or [-15,16], but it
can be set to [-2^(n-1)+1, 2^(n-1)].
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>POISMAP</B> <I>(series, sinfn, cosfn)</I>
<DD><A NAME="IDX437"></A>
will map the functions sinfn on the
sine terms and cosfn on the cosine terms of the poisson series given.
sinfn and cosfn are functions of two arguments which are a coefficient
and a trigonometric part of a term in series respectively.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>POISPLUS</B> <I>(A, B)</I>
<DD><A NAME="IDX438"></A>
is functionally identical to INTOPOIS(A+B).
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>POISSIMP</B> <I>(A)</I>
<DD><A NAME="IDX439"></A>
converts A into a Poisson series for A in general
representation.
</P>
</DL>
<P>
<DL>
<DT><U>special symbol:</U> <B>POISSON</B>
<DD><A NAME="IDX440"></A>
- The Symbol /P/ follows the line label of Poisson series
expressions.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>POISSUBST</B> <I>(A, B, C)</I>
<DD><A NAME="IDX441"></A>
substitutes A for B in C. C is a Poisson series.
(1) Where B is a variable U, V, W, X, Y, or Z then A must be an
expression linear in those variables (e.g. 6*U+4*V).
(2) Where B is other than those variables, then A must also be
free of those variables, and furthermore, free of sines or cosines.
POISSUBST(A, B, C, D, N) is a special type of substitution which
operates on A and B as in type (1) above, but where D is a Poisson
series, expands COS(D) and SIN(D) to order N so as to provide the
result of substituting A+D for B in C. The idea is that D is an
expansion in terms of a small parameter. For example,
POISSUBST(U,V,COS(V),E,3) results in COS(U)*(1-E^2/2) -
SIN(U)*(E-E^3/6).
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>POISTIMES</B> <I>(A, B)</I>
<DD><A NAME="IDX442"></A>
is functionally identical to INTOPOIS(A*B).
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>POISTRIM</B> <I>()</I>
<DD><A NAME="IDX443"></A>
is a reserved function name which (if the user has defined
it) gets applied during Poisson multiplication. It is a predicate
function of 6 arguments which are the coefficients of the U, V,..., Z
in a term. Terms for which POISTRIM is TRUE (for the coefficients of
that term) are eliminated during multiplication.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>PRINTPOIS</B> <I>(A)</I>
<DD><A NAME="IDX444"></A>
prints a Poisson series in a readable format. In common
with OUTOFPOIS, it will convert A into a Poisson encoding first, if
necessary.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>PSI</B> <I>(X)</I>
<DD><A NAME="IDX445"></A>
derivative of LOG(GAMMA(X)). At this time, MACSYMA does not
have numerical evaluation capabilities for PSI. For information on
the PSI[N](X) notation, see POLYGAMMA.
</P>
</DL>
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