 output.to from Sideway
Complex Analysis

Draft for Information Only

# Content

`Complex Function Complex Logarithm Function Examples Continuity of the Logarithm Function Derivative of Logarithm Function More General Theorem Application 1 Application 2 Terminology  Examples:`

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

# Complex Function

## Complex Logarithm Function

Given 𝑧∈ℂ\{0},, find 𝑤∈ℂ such that 𝑤=𝑧

𝑧=|𝑧|ℯ𝑖𝜃, then 𝑤=|𝑧|ℯ𝑖𝜃

Next, write 𝑤=𝑢+𝑖𝑣. Then 𝑢𝑖𝑣=|𝑧|ℯ𝑖𝜃

Thus 𝑢=|𝑧| and 𝑖𝑣=ℯ𝑖𝜃, so 𝑢=ln|𝑧| and 𝑣=𝜃+2𝑘𝜋=arg𝑧

Therefore, 𝑤=ln|𝑧|+𝑖arg𝑧

By definition. For 𝑧≠0,

`Log𝑧=ln|𝑧|+𝑖Arg 𝑧, the principal branch of logarithm,`

and

`log𝑧=ln|𝑧|+𝑖arg𝑧, a multi-valued function =Log𝑧+2𝑘𝜋𝑖, 𝑘∈ℤ`

## Examples

`Log𝑧=ln|𝑧|+𝑖Arg𝑧`
• Log1=ln|1|+𝑖Arg1=0
• Log𝑖=ln|𝑖|+𝑖Arg𝑖=0+𝑖𝜋2=𝑖𝜋2
• Log(-1)=ln|-1|+𝑖Arg-1=0+𝑖𝜋=𝑖𝜋
• Log(1+𝑖)=ln|1+𝑖|+𝑖Arg(1+𝑖)=ln2+𝑖𝜋4

## Continuity of the Logarithm Function

For the logarithm function, Log𝑧=ln|𝑧|+𝑖Arg𝑧

• z⟼|z| is continuous in
• z⟼ln|z| is continuous in ℂ\{0}
• z⟼Argz is continuous in ℂ\(−∞,0]
• Thus, Logz is continuous in ℂ\(−∞,0]
• However,
• as z→−𝑥∈(−∞,0) from above, Logz→ln𝑥+𝑖𝜋, and
• as z→−𝑥 from below, Logz→ln𝑥−𝑖𝜋,
so Logz is not continuous on (−∞,0) (and not defined at 0.

## Derivative of Logarithm Function

By Fact. The principal branch of logarithm, Logz, is analytic in ℂ\(−∞,0].

The derivative:

``` ℯLogz=z 𝑑𝑑z(ℯLogz)=𝑑𝑑zz ℯLogz⋅𝑑𝑑zLogz=1 𝑑𝑑zLogz=1ℯLogz=1z ```

## More General Theorem

By theorem. Suppose that 𝑓:𝑈→ℂ is an analytic function and there exists a continuous function 𝑔:𝐷→𝑈 from some domain 𝐷⊂ℂ into 𝑈 such that 𝑓(𝑔(z))=z for all z∈𝐷. Then 𝑔 is analytic in 𝐷, and

`𝑔′(z)=1𝑓′(𝑔(z)) for z∈𝐷`

## Application 1

Let 𝑓:ℂ→ℂ, 𝑓(z)=z2. Then 𝑓′(z)=2z.

Let 𝑔:ℂ\(−∞,0]→ℂ, 𝑔(z)=z be the principal branch of the square root (=|z|⋅ℯ𝑖Argz2). Then

• 𝑓(𝑔(z))=z for all z∈𝐷=ℂ\(−∞,0]
• 𝑔 is continuous in 𝐷, thus
• 𝑔 is analytic in 𝐷, and

```𝑔′(z)=1𝑓′(𝑔(z))  =12𝑔(z)  =12√2```

## Application 2

Let 𝑓:ℂ→ℂ, 𝑓(z)=z2. Then 𝑓′(z)=2z.

Let ℎ:ℂ\[0,∞)→ℂ, ℎ(z)={z, imz≥0z, imz<0 (=|z|⋅ℯ𝑖Argz2(+𝑖𝜋)|z|⋅ℯ𝑖Argz2|z|). Then

• 𝑓(ℎ(z))=z for all z∈𝐷=ℂ\[0,∞)
• is continuous in 𝐷, thus
• is analytic in 𝐷, and

```ℎ′(z)=1𝑓′(ℎ(z))  =12ℎ(z)```

## Terminology

Let 𝑓:𝑈→𝑉 be a function.

• 𝑓 is injective, also called 1-1, provided that 𝑓(𝑎)≠𝑓(𝑏) whenever 𝑎, 𝑏∈𝑈 with 𝑎≠𝑏.
• 𝑓 in surjective, also called onto, provided that for every 𝑦∈𝑉 there exists and 𝑥∈𝑈 such that 𝑓(𝑥)=𝑦.
• 𝑓 is a bijection, also called 1-1 and onto, if 𝑓 is both injective and surjective.

### Examples:

• 𝑓:{z∈ℂ|Rez>0}→ℂ\(−∞,0], 𝑓(z)=z2 is a bijection
• 𝑓:ℂ→ℂ, 𝑓(z)=z2 is not injective but is surjective.
• 𝑓:ℂ\(−∞,0]→ℂ, 𝑓(z)=z is injective but not surjective.

©sideway

ID: 190400013 Last Updated: 2019/4/13 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 (644) 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)

Latest Updated Links

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