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73. stirling


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73.1 Functions and Variables for stirling

Function: stirling (z,n)

Replace gamma(x) with the O(1/x^(2n-1)) Stirling formula. when n isn't a nonnegative integer, signal an error.

Reference: Abramowitz & Stegun, " Handbook of mathematical functions", 6.1.40.

Examples:

(%i1) load (stirling)$

(%i2) stirling(gamma(%alpha+x)/gamma(x),1);
       1/2 - x             x + %alpha - 1/2
(%o2) x        (x + %alpha)
                                   1           1
                            --------------- - ---- - %alpha
                            12 (x + %alpha)   12 x
                          %e
(%i3) taylor(%,x,inf,1);
                    %alpha       2    %alpha
          %alpha   x       %alpha  - x       %alpha
(%o3)/T/ x       + -------------------------------- + . . .
                                 2 x
(%i4) map('factor,%);
                                       %alpha - 1
         %alpha   (%alpha - 1) %alpha x
(%o4)   x       + -------------------------------
                                  2

The function stirling knows the difference between the variable 'gamma' and the function gamma:

(%i5) stirling(gamma + gamma(x),0);
                                    x - 1/2   - x
(%o5)    gamma + sqrt(2) sqrt(%pi) x        %e
(%i6) stirling(gamma(y) + gamma(x),0);
                         y - 1/2   - y
(%o6) sqrt(2) sqrt(%pi) y        %e
                                              x - 1/2   - x
                         + sqrt(2) sqrt(%pi) x        %e

To use this function write first load("stirling").


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This document was generated by Robert Dodier on December, 22 2007 using texi2html 1.76.