Chap 4

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Chap 4. Complex Algebra. For application to Laplace Transform Complex Number. Argand Diagram. y. r. q. x. Complex Variables. Continuous Function. Cplxdemo.m. Single Value Function. Many Values Function. Derivatives of Complex Variables. 1. 0. 0. 1. Cauchy Riemann Conditions.

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Chap 4

Complex Algebra

Complex Variables

Continuous Function

Cplxdemo.m

Single Value Function

Many Values Function

0

1

Cauchy Riemann Conditions

Analytic Functions

It has single value in the region R

It has a unique finite value

It has a unique finite derivative at z0,

satisfies the Cauchy Riemann Conditions

Example

Cauchy Riemann Conditions

Keep y constant

At Origin

One_OVER_Z.m

Singularities

Poles or unessential

Essential

Branch points

Poles or unessential Singularities

Second order Poles

Pole at a

Pole order p at zero

Pole order q at a

Branch Points

Many Value Function

Single

Cauchy’s Theorem

ถ้ามีฟังก์ชั่นใดที่เป็น Analytic ภายในหรือบน closed contour, integration รอบ contour จะได้ศูนย์

Stake’s theorem

Cauchy – Riemann conditions integral ทางด้านขวามือจะเป็นศูนย์

ตามเส้นทาง AB หรือ รอบเส้นทาง ACDB

path AB

curve ACDB

1. ตาม AC

2. เส้นโค้ง CDB ซึ่งมี constant radius 10

ผลรวมของ Integral

Example 2 Evaluate

around a circle with its center at the origin.

Although the function is not analytic function

Example 3 Evaluate

around a circle with its center at the origin.

This result is one of the fundamentals

of contour integration

Cauchy’s Integral formula

f(a) =constant at g

The theory of Residue

Pole at origin

Laurent expansion

Example 1 Evaluate

Around a circle center at the origin

if

Function is analytic

There is a pole order 3 at z = a

if

Evaluation without Laurent expansion

Many poles : independently evaluate

Example 2 Evaluate the residues of

Poles at 3,-4

Sum of Residues = 1

Example 4 evaluate

Around circle and

Pole at z = 0

Multiple Poles

Dividing throughout by