**Complex Function πcot(πz)/z ^{2}
**

**Eunju**** Sohn**

**The ****University**** of ****Illinois**** at ****Chicago**

MSGI Summer Workshop

2005

**Problem****:**

**integrate**** a complex function, πcot(πz)/z ^{2}
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/N ^{2} (N: integer) is π^{2}/6.**

**Poles and Residues when a contour C****5**** is taken**

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

**(Residue at the origin is -π ^{2}/3 not -π/3)**

The integral of modulus vs. N

**Visualization of the function ****πcot****(****πz****)/z ^{2}**

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 !!