output.to from Sideway
Complex Analysis

Draft for Information Only

# Content

`The Cauchy-Riemann equationsβComplex FunctionβCauchy-Riemann equations`

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

# The Cauchy-Riemann equations

## Complex Function

A complex function π can be written as π(π§)=π’(π₯,π¦)+ππ£(π₯,π¦) where π§=π₯+ππ¦ and π’, π£ are real-valued functions that depend on the two real variables π₯ and π¦.

Example:

`π(π§)=π§2=(π₯+ππ¦)2=π₯2-π¦2 π’(π₯,π¦) +πβ2π₯π¦ π£(π₯,π¦) `

Thus π’(π₯,π¦)=π₯2-π¦2 and π£(π₯,π¦)=2π₯π¦.

For example, if π§=2+π, then clearly π(π§)=(2+π)2=4+4π+π2=3+4π.

Alternatively, since π§=π₯+ππ¦, then π₯=2 and π¦=1, thus

`π’(π₯,π¦)=π’(2,1)=22-12=3 and π£(π₯,π¦)=π£(2,1)=2β2β1=4`

thus

`π(2+π)=π’(2,1)+ππ£(2,1)=3+4π`

Another example

`π(π§)=π§2=π’(π₯,π¦)+π=2π£(π₯,π¦), π’(π₯,π¦)=π₯2-π¦2, π£(π₯,π¦)=2π₯π¦`

Therefore

• π is differentiable everywhere in β
• πβ²(π§)=2π§ for all π§ββ

For the function π’(π₯,π¦)

• If fix the variable π¦ at a certain value, then π’ only depends on π₯. For example, if only consider π¦=3, then π’(π₯,π¦)=π’(π₯,3)=π₯2-9
• This function can now differentiate with respect to π₯ according to the rules of calculus and find that the derivative is 2π₯
• That is βοΏ½οΏ½οΏ½βπ₯(π₯,3)=𝑢π₯(π₯,3)=2π₯, and, more generally, for arbitrary (fixed) π¦, βπ’βπ₯(π₯,π¦)=π’π₯(π₯,π¦)=2π₯
• This is called the partial derivative of π’ with respect to π₯

Similarly, for the function π£(π₯,π¦)

• For example, π£(π₯,3)=2βπ₯β3=6π₯, and the derivative of this function with respect to π₯ is 6.
• Thus βπ£βπ₯(π₯,3)=π£π₯(π₯,3)=6
• More generally, for arbitrary (fixed) π¦, βπ£βπ₯(π₯,π¦)=π£π₯(π₯,π¦)=2π¦
• This is called the partial derivative of π£ with respect to π₯.

Obviously, the same thing can be done by fixing π₯ and differentiating with respect to π¦.

• Example: Let π₯ =2. Then π’(π₯,π¦)=π’(2,π¦)=4βπ¦2 and βπ’βπ¦(π₯,π¦)=π’π¦(2,π¦)=-2π¦
• More generally, βπ’βπ¦(π₯,π¦)=π’π¦(π₯,π¦)=-2π¦
• This is called the partial derivative of π’ with respect to π¦.
• Similarly, π£(2,π¦)=4π¦, and βπ£βπ¦(2,π¦)=π£π¦(2,π¦)=4. More generally, βπ£βπ¦(π₯,π¦)=π£π¦(π₯,π¦)=2π₯
• This is called the partial derivative of π£ with respect to π¦.

And the result are

`π(π§)=π§2=π’(π₯,π¦)+π=2π£(π₯,π¦), π’(π₯,π¦)=π₯2-π¦2, π£(π₯,οΏ½οΏ½οΏ½)=2π₯π¦`

The derivatives:

`πβ²(π§)=2π§, π’π₯(π₯,π¦)=2π₯, π’π¦(π₯,π¦)=-2π¦, π£π₯(π₯,π¦)=2π¦, π£π¦(π₯,π¦)=2x`

Notice:

• π’π₯=π£π¦
• π’π¦=-π£π₯
• πβ²=π’π₯+ππ£π₯ =ππ₯ =-π(π’π¦+ππ£π¦) =-πππ¦

Another Example

```π(π§)=2π§3-4π§+1, where π§=π₯+ππ¦  =2(π₯+ππ¦)3-4(π₯+ππ¦)+1  =2(π₯3+3π₯2ππ¦+3π₯π2π¦2+π3π¦3)-4π₯-4ππ¦+1  =(2π₯3-6π₯π¦2-4π₯+1) π’(π₯,π¦) +π(6π₯2π¦-2π¦3-4π¦) π£(π₯,π¦) ```

Then

```π’π₯(π₯,π¦)=6π₯2-6π¦2-4π£π₯(π₯,π¦)=12π₯π¦ π’π¦(π₯,π¦)=-12π₯π¦π£π¦(π₯,π¦)=6π₯2-6π¦2-4 ```

Thus, π’π₯=π£π¦ and π’π¦=-π£π₯

The derivatives:

```πβ²(π§)=6π§2-4, where π§=π₯+ππ¦ =6(π₯+ππ¦)2-4  =(6π₯2-6π¦2-4)+12ππ₯π¦  =π’π₯(π₯,π¦)+ππ£π₯(π₯,π¦)=-π(π’π¦(π₯,π¦)+ππ£π¦(π₯,π¦))=ππ₯(π§)=-πππ¦(π§) ```

## Cauchy-Riemann equations

By Theorem. Suppose that π(π§)=π’(π₯,π¦)+ππ£(π₯,π¦) is differentiable at a point π§0. Then the partial derivatives π’π₯, π’π¦, π£π₯, π£π¦ exist at π§0, and satisfy there:

`π’π₯=π£π¦ and π’π¦=-π£π₯`

These are called the Cauchy-Riemann Equations. Also,

```πβ²(π§0)=π’π₯(π₯0,π¦0)+ππ£π₯(π₯0,π¦0)=ππ₯(π§0)  =-π(π’π¦(π₯0,π¦0)+ππ£π¦(π₯0,π¦0))=-πππ¦(π§0)```

Method of Proof

For the difference quotient,

`π(π§0+β)-π(π§0)β`

whose limit as ββ0 must exist if π is differentiable at π§0

• Let β approach 0 along the real axis only first, and then along the imaginary axis only next. Both times the limit must exist and both limits must be the same.
• Equating these two limits with each other and recognizing the partial derivatives in the expressions yields the Cauchy-Riemann equations.

Another example

Let π(π§)=π§=π₯-ππ¦, then π’(π₯,π¦)=π₯ and π£(π₯,π¦)=-π¦, so

```π’π₯(π₯,π¦)=1π£π₯(π₯,π¦)=0 π’π¦(π₯,π¦)=0π£π¦(π₯,π¦)=-1 ```

Since π’π₯(π₯,π¦)β π£π¦(π₯,π¦) (while π’π¦(π₯,π¦)=-π£π₯(π₯,π¦) for all π§, the function π is not differentiable anywhere.

Recall: If π is differentiable at π§0 then the Cauchy-Riemann equations hold at π§0. However, if π satisfies the Cauchy-Riemann equations at a point π§0 then does this imply that π is differentiable at π§0? And what is the sufficient conditions for differentiability.

By theorem. Let π=π’+ππ£ be defined on a domain π·ββ. Then π is analytic in >π· if and only if π’(π₯,π¦) and π£(π₯,π¦) have continuous first partial derivatives on >π· that satisfy the Cauchy-Riemann equations.

Example: π(π§)=ππ₯cosπ¦+πππ₯sinπ¦. The

```π’π₯(π₯,π¦)=ππ₯cosπ¦π£π₯(π₯,π¦)=ππ₯sinπ¦ π’π¦(π₯,π¦)=βππ₯sinπ¦π£π¦(π₯,π¦)=ππ₯cosπ¦ ```

Thus the Cauchy-Riemann equations are satisfied, and in addition, the functions π’π₯, π’π¦, π£π₯, π£π¦ are continuous in β. Therefore, the function π is analytic in β, thus entire.

Β©sideway

ID: 190300030 Last Updated: 2019/3/30 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