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The Maxima on-line user's manual

Algebra Calculator

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Gamma Calculator

Gamma

Function: gamma (<z>) The basic definition of the gamma function (A&S 6.1.1) is

g:4;
hg:4;
e:sqrt((g/2)^2+hg^2),numer;
f:sqrt((g/2)^2+hg^2);
alpha:atan(h/(g/2))*180/%pi,numer;
beta:asin((g/2)/e)*180/%pi,numer;
gamma:180-alpha-beta;
U:e+f+g;
s:U/2.0;
F:sqrt(s*(s-e)*(s-f)*(s-g));
F:g*h;

inf / [ z - 1 - t gamma(z) = I t %e dt ] / 0

Maxima simplifies gamma for positive integer and positive and negative rational numbers. For half integral values the result is

     a rational number times sqrt(%pi). The simplification for
     integer values is controlled by factlim. For integers greater
     than factlim the numerical result of the factorial function,
     which is used to calculate gamma, will overflow.  The
     simplification for rational numbers is controlled by gammalim to
     avoid internal overflow. See factlim and gammalim.

For negative integers gamma is not definied.

Maxima can evalute gamma numerically for real and complex values in float and bigfloat precision.

gamma has mirror symmetry.

When gamma_expand is true, Maxima expands gamma for arguments z+n and z-n where n is an integer.

Maxima knows the derivate of gamma.

Examples:

Simplification for integer, half integral, and rational numbers:

          (%i1) map(gamma,[1,2,3,4,5,6,7,8,9]);
          (%o1)               [1, 1, 2, 6, 24, 120, 720, 5040, 40320]

          (%i2) map(gamma,[1/2,3/2,5/2,7/2]);
                                     sqrt(%pi)  3 sqrt(%pi)  15 sqrt(%pi)
          (%o2)          [sqrt(%pi), ---------, -----------, ------------]
                                         2           4            8

          (%i3) map(gamma,[2/3,5/3,7/3]);
                                                  2           1
                                          2 gamma(-)  4 gamma(-)
                                      2           3           3
          (%o3)                [gamma(-), ----------, ----------]
                                      3       3           9

Numerical evaluation for real and complex values:

          (%i4) map(gamma,[2.5,2.5b0]);
          (%o4)             [1.329340388179137, 1.329340388179137b0]

          (%i5) map(gamma,[1.0+%i,1.0b0+%i]);
          (%o5) [.4980156681183558 - .1549498283018108 %i,
                                4.980156681183561b-1 - 1.549498283018107b-1 %i]

gamma has mirror symmetry:

          (%i8) declare(z,complex)$
          (%i9) conjugate(gamma(z));
          (%o9)                         gamma(conjugate(z))

Maxima expands gamma(z+n) and gamma(z-n), when gamma_expand is true:

          (%i10) gamma_expand:true$
          (%i11) [gamma(z+1),gamma(z-1),gamma(z+2)/gamma(z+1)];
                                                gamma(z)
          (%o11)                   [z gamma(z), --------, z + 1]
                                                 z - 1

The deriviative of gamma:

          (%i12) diff(gamma(z),z);
          (%o12)                         psi (z) gamma(z)
                                            0

See also makegamma.

The Euler-Mascheroni constant is %gamma.

There are also some inexact matches for gamma. Try ?? gamma to see them.

(%o1)                                true
(%i2) 

Gamma Example

Related Examples

gamma-linsolve

eq1:x*4+y-2=y;

eq2:x+2=y;

eq3: pi*(1+(x-((a-1)/...

Calculate

gamma-inf-limit

limit (((1/x)^(72 + 1...

Calculate

gamma-inf-limit

limit ((gamma(x + 72 ...

Calculate

gamma

eq1:x=lambda+mu+gamma;

eq2:y=lambda-mu+3*gamma;

eq3:z=lambda+2*mu;

Calculate

gamma-solve

eq1:(2*pi*ft/gamma)^2...

eq2:2*pi*fn/gamma=Heo...

solve([eq1,eq2],[gamm...

Calculate

gamma-inf-limit-log

limit ((gamma(x + 72 ...

Calculate

gamma-numer

gamma(2),numer;

Calculate

gamma-log-pi-plot2d
plot2d([pi(x) / log(pi(x))], [x, 2, 10]);

pi(x) := gamma(x + 1);

plot2d([pi(x) / log(p...

Calculate

gamma-sqrt

_c : 299792458 /* met...

_gamma(v) := 1 / sqrt...

ke_rel(m_0, v) := (_g...

Calculate

gamma-taylor

taylor(%pi^(-s)*gamma...

Calculate