output.to from Sideway
Complex Analysis

Draft for Information Only

# Content

`Complex FunctionβComplex Trigonometric FunctionβProperties of Sine and Cosine`

source/reference:

# Complex Function

## Complex Trigonometric Function

For the function, β―ππ=cosπ+πsinπ

```Let β―ππ=cosπ+πsinπ Therefore, β―βππ=cos(βπ)+πsin(βπ)=cosπβπsinπ Hence, β―ππ+β―βππ=2cosπ and β―ππββ―βππ=2πsinπ Thus cosπ=β―ππ+β―βππ2 and sinπ=β―ππββ―βππ2π```

By definition, The complex cosine and sine functions are defined via

`cosπ§=β―ππ§+β―βππ§2 and sinπ§=β―ππ§ββ―βππ§2π`

## Properties of Sine and Cosine

For the function, cosπ§=β―ππ§+β―βππ§2 and sinπ§=β―ππ§ββ―βππ§2π

• cosπ§ and sinπ§ are analytic functions (in fact, entire).
• For real-valued π§, (i.e. π§=π₯+πβ0) the complex sine and cosine agree with the real-valued sine and cosine functions.
• cos(βπ§)=β―βππ§+β―ππ§2=cosπ§
• sin(βπ§)=β―βππ§ββ―ππ§2π=βsin𝑧
• cos(π§+π€)=cosπ§cosπ€βsinπ§sinπ€, sin(π§+π€)=sinπ§cosπ€+cosπ§sinπ€

Proofs of the addition formulae cos(π§+π€)

``` cosπ§cosπ€βsinπ§sinπ€ =(β―ππ§+β―βππ§2)(β―ππ€+β―βππ€2) β(β―ππ§ββ―βππ§2π)(β―ππ€ββ―βππ€2π) =(β―ππ§+β―βππ§)(β―ππ€+β―βππ€)+(β―ππ§ββ―βππ§)(β―ππ€ββ―βππ€)4 =β―ππ§β―ππ€+β―ππ§β―βππ€+β―βππ§β―ππ€+β―βππ§β―βππ€+β―ππ§β―ππ€ββ―ππ§β―βππ€ββ―βππ§β―ππ€+β―βππ§β―βππ€4 =2β―ππ§β―ππ€+2β―βππ§β―βππ€4 =β―π(π§+π€)+β―βπ(π§+π€)2 =cos(π§+π€)```
• cos(π§+2π)=β―π(π§+2π)+β―βπ(π§+2π)2=cosπ§
• sin(π§+2π)=β―π(π§+2π)ββ―βπ(π§+2π)2π=sinπ§
• cos2π§+cos2π§=1. Proof: Let π€=βπ§ in the addition formula for cosine.
• sin(π§+π2)=cosπ§

Proof:

```sin(π§+π2)=β―π(π§+π2)ββ―βπ(π§+π2)2π  =πβ―ππ§β(βπ)β―βππ§2π  =β―ππ§+β―βππ§2=cosπ§ ```
• sinπ§=0βπ§=ππ, πββ€

Proof

```sinπ§=0ββ―ππ§ββ―βππ§2π=0  ββ―ππ§=β―βππ§  βππ§β(βππ§)=2πππ, πββ€, the periodicity of the exponential with period of 2πππ.  β2ππ§=2πππ, πββ€  βπ§=ππ, πββ€ ```
• cosπ§=0βπ§=π2+ππ, πββ€

Proof

```cosπ§=0ββ―ππ§+β―βππ§2=0  ββ―2ππ§+12=(β―ππ§+π)(β―ππ§βπ)2=0  ββ―2ππ§+1=0  ββ―2ππ§=β1=β―ππ  β2ππ§βππ=2πππ, πββ€, the periodicity of the exponential with period of 2πππ  β2ππ§=(2π+1)ππ, πββ€  βπ§=π2+ππ, πββ€ ```
• Derivative of Sine: πππ§sinπ§=cosπ§

Proof

```πππ§sinπ§=πππ§β―ππ§ββ―βππ§2π  =πβ―ππ§β(βπ)β―βππ§2π  =β―ππ§+β―βππ§2=cosπ§ ```
• Derivative of Cosine: `πππ§cosπ§=βsinπ§`

Proof

```πππ§cosπ§=πππ§β―ππ§+β―βππ§2  =πβ―ππ§+(βπ)β―βππ§2  =π(β―ππ§ββ―βππ§)2=βsinπ§ ```
• Complex sine in terms of real functions, `sinπ§=sinπ₯coshπ¦+πcosπ₯sinhπ¦`

Proof

```sinπ§=sin(π₯+ππ¦)  =sinπ₯cos(ππ¦)+cosπ₯sin(ππ¦)  =sinπ₯β―π(ππ¦)+β―βπ(ππ¦)2+cosπ₯β―π(ππ¦)ββ―βπ(ππ¦)2π  =sinπ₯β―βπ¦+β―π¦2+cosπ₯β―βπ¦ββ―π¦2π  =sinπ₯β―π¦+β―βπ¦2+πcosπ₯β―π¦ββ―βπ¦2  =sinπ₯coshπ¦+πcosπ₯sinhπ¦ ```
• Complex cosine in terms of real functions, `cosπ§=cosπ₯coshπ¦+πsinπ₯sinhπ¦`

Proof

```cosπ§=cos(π₯+ππ¦)  =cosπ₯cos(ππ¦)βsinπ₯sin(ππ¦)  =cosπ₯β―π(ππ¦)+β―βπ(ππ¦)2+sinπ₯β―π(ππ¦)ββ―βπ(ππ¦)2π  =cosπ₯β―βπ¦+β―π¦2+sinπ₯β―βπ¦ββ―π¦2π  =cosπ₯β―π¦+β―βπ¦2+πsinπ₯β―π¦ββ―βπ¦2  =cosπ₯coshπ¦+πsinπ₯sinhπ¦ ```

ID: 190400011 Last Updated: 2019/4/11 Revision:

Home (5)

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)