%  DO YOU BELIEVE ...
%
%    the Cauchy integral formula:  Example (14.69), p. 14-14?
%
%    Taylor and Laurent series:  Example (15.51), p. 15-8?
%

%  Let's check formula (14.71), p. 14-14.

N = 400;
z = 3*exp(1i*linspace(0,2*pi,N)');   % Points around circle of radius 3 about 0,
                                     % oriented counterclockwise since theta runs
                                     % from 0 to 2 pi.

I = 0;                               % I will approximate the integral (14.69)
for j=2:2:N,
  fofz = exp(-z(j))/(z(j)+2);        % Evaluate f at midpoint of interval
  jp1 = j+1;                         % from z(j-1) to z(j+1).
  if j==N, jp1 = 1; end;
  dz = z(jp1) - z(j-1);              % Delta z
  I = I + fofz*dz;                   % Add f(z)*(Delta z).
end;
I, Cauchyvalue = 2*pi*1i*exp(2), diff=Cauchyvalue-I, pause,

%  Let's check formula (15.51), p. 15-8.

zm1 = 4*rand(1)*exp(1i*2*pi*rand(1));% Generate random value for z-1,
                                     % with absolute value beween 0 and 4.
z = zm1 + 1;
fofz = -4/((z+3)*(z-1));             % f in (15.39)
series = -1/(z-1);                   % series will approximate (15.51)
for j=1:100,
  series = series + (-1)^(j-1)*(z-1)^(j-1)/4^j;
end;
fofz, series, diff=fofz-series