## Complex #27: integrating over a long rectangle

**Problem 27: Show that has the same value on every straight line path (oriented in the direction) parallel to the real axis.**

Proof: Consider the rectangle , where is some given number and is a very large number. Since the integrand is entire, the integral over the boundary of is zero by the Cauchy Integral Theorem, and as , the horizontal pieces correspond to with being the line or the line , with one being negative and the other positive. Thus, the result is proved if we can show that the contributions from the vertical pieces go to zero as goes to .

Observe that the absolute value of the integrand at is , which on each vertical boundary piece is for . Integrating this over will give a value bounded by a constant times , which clearly goes to zero as .

QED.