Let f be entire. Evaluate the integral from zero to 2 pi of f(z_0+re^(i theta)) e^(ik theta), where z_0 is a constant and k is a constant greater than or equal to 1.
Cauchy's theorem tells us that the integral of an entire function around a closed contour is zero. Attached is a 1/2 page solution written in Word with equations in Mathtype illustrating the use of this theorem. It also illustrates how substitution is often used to turn an integral into a contour integral to which the theorem may be applied.