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# 72. stirling

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## 72.1 Functions and Variables for stirling

Function: stirling (z, n)
Function: stirling (z, n, pred)

  Replace `gamma(x)` with the O(1/x^(2n-1)) Stirling formula. When n isn't a nonnegative integer, signal an error. With the optional third argument `pred`, the Stirling formula is applied only when `pred` is true.

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

  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 apply the Stirling formula only to terms that involve the variable `k`, use an optional third argument; for example

      (%i7) makegamma(pochhammer(a,k)/pochhammer(b,k));
                            gamma(b) gamma(k + a)
      (%o7)                 ---------------------
                            gamma(a) gamma(k + b)

      (%i8) stirling(%,1, lambda([s], not(freeof(k,s))));
              b - a                 k + a - 1/2        - k - b + 1/2
            %e      gamma(b) (k + a)            (k + b)
      (%o8) --------------------------------------------------------
                                    gamma(a)

  The terms `gamma(a)` and `gamma(b)` are free of `k`, so the Stirling formula was not applied to these two terms.

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