# MATH3401

Lecture 1

## Complex Analysis

• $$e^{i\pi} = -1$$
• $$\ds \int_0^{\infty}\dfrac{\sin x}{x} dx=\dfrac{\pi}{2}$$
• Recall that $$\ds \int_0^1\dfrac{dx}{x}= \ds \lim_{\varepsilon \ra 0^+}\int_{\varepsilon}^1\dfrac{dx}{x}$$ diverges
• And $$\ds \int_1^{\infty}\dfrac{dx}{x}= \ds \lim_{M \ra \infty}\int_{1}^{M}\dfrac{dx}{x}$$ also diverges

## Complex Analysis

Methods for contour integrals

Fluid flow

Air flow

## Complex Analysis

Series

$$\text{I.} \;\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots =\frac{\pi^2}{6}$$

$$\text{II.} \;\frac{1}{1^2}-\frac{1}{2^2}+\frac{1}{3^2}-\cdots =\frac{\pi^2}{12}$$

$$\text{III.} \;{\small \ds \sum_{k=1}^{\infty} \frac{1}{1+4k^2\pi^2}=\frac{1}{2}\left( \frac{1}{e-1}-\frac{1}{2}\right)}$$

## Complex Analysis

Series

$$\zeta(s) = \ds \sum_{n=1}^{\infty}\frac{1}{n^s}= \ds \prod_{p\;\text{prime}}(1-p^{-s})^{-1}$$

$$\zeta$$ is the Riemann zeta function.

## Complex Analysis

Riemann hypothesis

$$\zeta(s) = \ds \sum_{n=1}^{\infty}\frac{1}{n^s}= \ds \prod_{p\;\text{prime}}(1-p^{-s})^{-1}$$

$$\zeta$$ has infinitely many non-trivial zeros and they all lie on the line $$\Re(s)=\dfrac{1}{2}$$

## Complex Analysis

Riemann hypothesis

$$\zeta(s) = \ds \sum_{n=1}^{\infty}\frac{1}{n^s}= \ds \prod_{p\;\text{prime}}(1-p^{-s})^{-1}$$

Only makes sense for $$\Re(s)>1$$:

Need to extend $$\zeta$$ to $$\C$$ via analytic continuation.

## Complex Analysis

Riemann hypothesis

$$\zeta(s) = \ds \sum_{n=1}^{\infty}\frac{1}{n^s}= \ds \prod_{p\;\text{prime}}(1-p^{-s})^{-1}$$

Trivial zeros are $$-2, -4, -6, \cdots$$

Millenium problem!