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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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