The gamma function and its poles

Nick Hale, December 2009

This script displays some of the features introduced in version 3 for unbounded functions by demonstrating with the gamma function on the interval [-4 4].

The gamma function on [-4 4] has simple poles at the negative integers and zero. Chebfun can determine the locations and orders of these poles if it is called with the 'blowup' and 'splitting' flags on. The 'exponents' field of the output indicates that each pole is simple, that is, it has a singularity of type x^(-1).

LW = 'linewidth'; FS = 'fontsize'; MS = 'markersize';
gam = chebfun('gamma(x)',[-4 4],'blowup','on','splitting','on')
plot(gam,'b',LW,1.6), hold on
title('Gamma function',FS,16)

gam = 
   chebfun column (5 smooth pieces)
       interval       length   endpoint values   exponents 
[      -4,      -3]       23      Inf      Inf   -1   -1   
[      -3,      -2]       22     -Inf     -Inf   -1   -1   
[      -2,      -1]       22      Inf      Inf   -1   -1   
[      -1,-2.3e-308]       20     -Inf     -Inf   -1   -1   
[-2.3e-308,       4]       35      Inf        6   -1    0   
Total length = 122   vertical scale =  12

We can now treat the gamma function like any other chebfun. For example, we can:

(1) Find its reciprocal

gam_i = 1./gam;

(2) Compute the square root of abs(gamma)

absgam = abs(gam);
sqrtgam = real(sqrt(absgam));

(3) Plot these functions

plot(gam_i,'r', sqrtgam,'-g',LW,1.6)
legend('\Gamma(x)', '1/\Gamma(x)', 'sqrt(|\Gamma(x)|)')
title('Various related functions',FS,16)

(4) Plot the critical points

[y r] = minandmax(gam,'local');
[yi ri] = minandmax(gam_i,'local');
[ys rs] = minandmax(sqrtgam,'local');

plot(r,gam(r),'.k',ri,gam_i(ri),'.k', ...
    rs,sqrtgam(rs),'.k',MS,18,LW,1.6), hold off
title('Gamma function on [-4 4] and its critical points',LW,16)

(5) Copute some integrals

sum(gam)
sum(absgam)
sum(sqrtgam)

ans =
   NaN
ans =
   Inf
ans =
  14.043323986892389