Suppose you only have a real integral numerical calculator, how would you use it to verify complex integrals numerically?
Well, of course you can use Residue theorem, Cauchy’s integral formula, etc. to theoretically compute integrals. However, I’m often interested that if there are any methods to verify it, other than double checking my work.
One possible way is seperating the real part and the imaginary part of the integrand. However, that could be cumbersome. Consider the integral $$\oint_{|z|=10}\frac1{(z-1)(z-2)(z-3)(z-4)(z-5)}dz$$ I cannot imagine what its real and imaginary parts could be, but I know it equals $$2\pi i(\frac1{24}-\frac16+\frac14-\frac16+\frac1{24})$$
Thanks in advance.