Sideway
output.to from Sideway
Complex Analysis



Draft for Information Only

Content

Complex Function
  Function
 Complex Function
  Graphing Complex Function

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

Complex Function

Function

In general, a function ƒ:A→B is a rule that assigns to each element of A exactly one element of B. for example, ƒ:ℝ→ℝ, ƒ(x)=x²+1. Since ℝ can be represented by a one dimensional number line, the function mapping can be represented by a graph in the x-y plane.

Complex Function

For a complex function, for example, ƒ:ℂ→ℂ ƒ(z)=z²+1.

Let z=x+𝑖y, w=ƒ(z)
⇒w=ƒ(z)=(x+𝑖y)²+1
⇒w=(x²-y²+1)+𝑖(2xy)
⇒w=u(x,y)+𝑖v(x,y) where u,v:ℝ²→ℝ

Graphing Complex Function

Two complex planes are used to graph the complex function. One is for the domain and one is for the range. These complex planes can be used to analyze how geometric configurations in the z-plane are mapped under ƒ to the w-plane.

For example

Let w=ƒ(z)=z² , z=re𝑖θ
⇒w=(re𝑖θ)²=r²e2𝑖θ
⇒|w|=|z|², arg w=2arg z

As z moves around a circle of radius r in z-plane once, w moves around the circle of radius r² at double speed twice in w-plane.

For example

Let w=w'+c, c∊ℂ, w'=ƒ(z)=z²,  z=re𝑖θ
⇒w'=(re𝑖θ)²=r²e2𝑖θ
⇒|w'|=|z|², arg w'=2arg z
∴w=r²e2𝑖θ+c

©sideway

ID: 190300017 Last Updated: 2019/3/17 Revision:

IMAGE

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)new

CSS (SC)

ASP.NET (SC)

Regular Expression (SC)

HTML

Knowledge Base

Common Color (SC)

Html Entity (Unicode) (SC)

Html 401 Special (SC)

OS (389)new

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)

Fluid Kinematics (5)

Control

Process Control (1)

Acoustics (19)

FiniteElement (2)

Biology (1)

Geography (1)

Latest Updated Links

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