# User:Caliburn

b. 04/03/2001 from Kent.

My interest currently lies in complex analysis, which I've been studying using Gamelin, though I intend to broaden this.

2019 Cambridge applicant. (Queens')

I prefer to be called by my real name, George, as opposed to Caliburn or any variation.

## To do

- Hankel Representation of Riemann Zeta Function - proved that it agrees for $\Re \paren z > 1$, yet to prove that it admits an analytic continuation. Deduce Functional Equation for Riemann Zeta Function and Riemann Zeta Function at Negative Integers.
- Sum of Reciprocals of Squares of Odd Integers via $\displaystyle \sum_{n \mathop = -\infty}^\infty \frac 1 {\paren {z - \frac 1 2}^2}$ and Basel Problem with the summation formula.
- Give more general proof for Residue Theorem.
- $\displaystyle \sum_{n \mathop = -\infty}^\infty \frac 1 {n^2 + a^2} = \frac \pi a \coth \paren {\pi a}$ via summation formula, implies $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^2} = \lim_{a \to 0} \paren {\frac \pi {2 a} \coth \paren {\pi a} - \frac 1 {2 a^2} } = \frac {\pi^2} 6$. Not sure what to call this.
- Use cosec instead to give an adaption of summation formula for series of the form $\sum_{n \mathop \in \Z} \paren {-1}^n f \paren n$.
- Adapt previous contributions to account for new macros.