## Coursera: analysis complex kind Week 3.4. Complex trig functions.

### Aim

To define sin(z), cos(z), and introduce sinh(z) and cosh(z), and relationships between these functions, where z $\in \mathbb{C}$

### Introduction

since $e^{i\theta} = cos(\theta) + i sin(\theta)$ and hence

$e^{-i\theta} = cos(\theta) - i sin(\theta)$

combining: cos z = $\frac{e^{iz}+ e^{-iz}}{2}$ and sin z = $\frac{e^{iz}- e^{-iz}}{2i}$

usual properties of sin and cos apply and identities cos(z + w) and sin (z + w) apply (expand and simplify cos (x + iy + u + iv) etc.)

Use definitions of cos(z) and sin(z) to show cos (z + 2π) = cos (z) and sin (z + 2π) = sin (z)

so periodicity remains as 2π. Compare to period of e^z.

### Slide 6

When is sin z = 0?

$0=e^{iz}- e^{-iz}$, since only numerator relevant in making zero.

$e^{iz} = e^{-iz}$

iz -(-iz) = 2ikπ  ** see definition of exponential function to explain this **

2iz = 2ikπ so, z = kπ, this is a real number.

So sin z = 0 only for (real) z = kπ ***

Similarly cos z = 0 only for (real) z = π/2 + kπ ***(b)

### Slide 7

Derivative sin (z) and cos(z)

Method: Use definitions of sin and cos in terms of exponential functions then apply the chain rule when differentiating to show

d/dz(sin z) = cos z

d/dz(cos z) = -sin z