Complex Function πcot(πz)/z²

Eunju Sohn

The University of Illinois at Chicago

MSGI Summer Workshop

2005

Reed College

Problem:

integrate a complex function, πcot(πz)/z² over a specific contour where the function has no singularities.

Goal:

if we take a contour large enough, we can show that the modulus of the integrand goes to zero and by Residue Theorem, can prove the infinite sum of 1/N² (N: integer) is π²/6.

Poles and Residues when a contour C5 is taken

( CN : N = 5, x = +,- 5.5, y = +,- 5.5)

(Residue at the origin is -π²/3 not -π/3)

The integral of modulus vs. N

Visualization of the function πcot(πz)/z²

Both pictures show changes in the arguments of the function

Visualization of Some Complex Function

Z + 1/Z

Both pictures show changes in the arguments of the function

The modulus of cot(πz)

The visualization of the modulus of the complex function cot(πz)

The area where the modulus of the function is bigger than one is painted as black.

Picture resulted by dividing the modulus by 2

The picture shows cot(πz) is bounded on the complex plane except at its poles

Big, big thank to

the organizers of the workshop: David Austin, Jim Fix , and Bill Casselman

and

Jiho Kim who helped me a lot with java programing !!