 output.to from Sideway
Complex Analysis

Draft for Information Only

# Content

`Complex Function Complex Exponential Function Properties`

source/reference:
https://www.youtube.com/channel/UCaTLkDn9_1Wy5TRYfVULYUw/playlists

# Complex Function

## Complex Exponential Function

For the function, 𝑓(𝑧)=ℯ𝑥cos𝑦+𝑖ℯ𝑥sin𝑦, (where 𝑧=𝑥+𝑖𝑦) is an entire (=analytic in function.

Some of its properties:

• if 𝑦=0, then 𝑓(𝑧)=𝑓(𝑥+𝑖⋅0)=𝑓(𝑥)=ℯ𝑥, so 𝑓 agrees with the "regular" exponential function on ℝ
• 𝑓(𝑧)=ℯ𝑥(cos𝑦+𝑖sin𝑦)=ℯ𝑥𝑖𝑦

By definition. The complex exponential function, 𝑧, sometimes also denoted exp(𝑧), is defined by

`ℯ𝑧=ℯ𝑥⋅ℯ𝑖𝑦, where 𝑧=𝑥+𝑖𝑦`

## Properties

For the function, 𝑧= ℯ𝑥⋅ℯ𝑖𝑦, where 𝑧=𝑥+𝑖𝑦:

• |ℯ𝑧|=|ℯ𝑥||ℯ𝑖𝑦|=ℯ𝑥
• arg𝑧=arg(ℯ𝑥𝑖𝑦)=𝑦(+2𝜋𝑘, where 𝑘∈ℤ)
• 𝑧+2𝜋𝑖=ℯ𝑥𝑖(𝑦+2𝜋)=ℯ𝑥𝑖𝑦=ℯ𝑧
• ```ℯ𝑧+𝑤=ℯ(𝑥+𝑖𝑦)+(𝑢+𝑖𝑣), where 𝑧=𝑥+𝑖𝑦, 𝑤=𝑢+𝑖𝑣  =ℯ(𝑥+𝑢)+𝑖(𝑦+𝑣)=ℯ𝑥ℯ𝑢ℯ𝑖𝑦ℯ𝑖𝑦  =(ℯ𝑥ℯ𝑖𝑦)(ℯ𝑢ℯ𝑖𝑦)=ℯ𝑧ℯ𝑤```
• 1𝑧=ℯ−𝑧, since ℯ𝑧−𝑧=ℯ0=1
• 𝑧 is an entire function.
• Derivative 𝑓′(𝑧):

Let 𝑢(𝑥,𝑦)=ℯ𝑥cos𝑦, 𝑣(𝑥,𝑦)=ℯ𝑥sin𝑦

Then ```𝑢𝑥(𝑥,𝑦)=𝑒𝑥cos𝑦;𝑣𝑥(𝑥,𝑦)=𝑒𝑥sin𝑦 𝑢𝑦(𝑥,𝑦)=−𝑒𝑥sin𝑦;𝑣𝑦(𝑥,𝑦)=𝑒𝑥cos𝑦```

Thus 𝑓′(𝑧)=𝑢(𝑥,𝑦)+𝑖𝑣(𝑥,𝑦)=ℯ𝑥cos𝑦+𝑖ℯ𝑥sin𝑦=ℯ𝑧

So the derivative of 𝑧 is 𝑧, in symbols, `dd𝑧ℯ𝑧=ℯ𝑧`.

• `dd𝑧ℯ𝑎𝑧=𝑎⋅ℯ𝑎𝑧 (𝑎∈ℂ)` by the chain rule
• 𝑧=ℯ𝑥−𝑖𝑦=ℯ𝑥−𝑖𝑦=ℯ𝑥𝑖𝑦=𝑥𝑖𝑦=𝑧
• 𝑧=1 if and only if 𝑥𝑖𝑦=1. The complex number in polar form, 𝑥𝑖𝑦, equals 1, when its length equals 1 and its argument equals 0, ie.e. when 𝑥 and y=2𝑘𝜋, 𝑘∈ℤ. Thus

`ℯ𝑧=1⇔𝑧=2𝜋𝑖𝑘, 𝑘∈ℤ`
• 𝑧=ℯ𝑤⇔ℯ𝑧−𝑤=1⇔𝑧−𝑤=2𝜋𝑖𝑘⇔𝑧=𝑤+2𝜋𝑖𝑘

The function 𝑤=ℯ𝑧 is a mapping from `ℂ 𝑧-plane ` to `ℂ 𝑤-plane `.

For the images of horizontal lines, 𝐿={𝑥+𝑖𝑦0|𝑥∈ℝ} for fixed 𝑦0∈ℝ. Then 𝑧=ℯ𝑥+𝑖𝑦0=ℯ𝑥𝑖𝑦0, a line from origin but not equal with fixed angle.

For the images of vertical lines, 𝐿={𝑥0+𝑖𝑦|𝑦∈ℝ} for fixed 𝑥0∈ℝ. Then 𝑧=ℯ𝑥0+𝑖𝑦=ℯ𝑥0𝑖𝑦, a circle with center at origin.

For the images of vertical strip, 𝑆={𝑧:0<Re𝑧<1}, a ring between circle of value 0 and e

• When 𝑧=0
```ℯ𝑧=0⇔ℯ𝑥⋅ℯ𝑖𝑦=0 Note: ℯ𝑖𝑦 has absolute value 1  ⇔ℯ𝑥=0  ⇔Never...!```
• For a given 𝑧∈ℂ\{0}, is there a 𝑤∈ℂ such that 𝑤=𝑧? Writing 𝑧=|𝑧|ℯ𝑖𝜃 and 𝑤=𝑢+𝑖𝑣 this is equivalent to:
```ℯ𝑤=𝑧⇔ℯ𝑢ℯ𝑖𝑣=|𝑧|ℯ𝑖𝜃  ⇔ℯ𝑢=|𝑧| and ℯ𝑖𝑣=ℯ𝑖𝜃  ⇔𝑢=ln|𝑧| and 𝑣=𝜃+2𝑘𝜋  ⇔𝑤=ln|𝑧|+𝑖arg𝑧```

This is the complex logarithm.

©sideway

ID: 190400003 Last Updated: 2019/4/3 Revision: Home (5)

Business

Management

HBR (3)

Information

Recreation

Hobbies (7)

Culture

Chinese (1097)

English (336)

Reference (66)

Computer

Hardware (149)

Software

Application (187)

Digitization (24)

Numeric (19)

Programming

Web (648) CSS (SC)

ASP.NET (SC)

HTML

Knowledge Base

Common Color (SC)

Html 401 Special (SC)

OS (389)

MS Windows

Windows10 (SC)

.NET Framework (SC)

DeskTop (7)

Knowledge

Mathematics

Formulas (8)

Number Theory (206)

Algebra (20)

Trigonometry (18)

Geometry (18)

Calculus (67)

Complex Analysis (21)

Engineering

Tables (8)

Mechanical

Mechanics (1)

Rigid Bodies

Statics (92)

Dynamics (37)

Fluid (5)

Control

Acoustics (19)

Biology (1)

Geography (1)

Copyright © 2000-2019 Sideway . All rights reserved Disclaimers last modified on 10 Feb 2019