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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

Chap 4

Complex Algebra

slide5

Complex Variables

Continuous Function

Cplxdemo.m

slide6

Single Value Function

Many Values Function

slide8

0

1

Cauchy Riemann Conditions

slide9

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

slide11

Example

Cauchy Riemann Conditions

slide12

Keep y constant

At Origin

One_OVER_Z.m

slide13

Singularities

Poles or unessential

Essential

Branch points

slide14

Poles or unessential Singularities

Second order Poles

Pole at a

Pole order p at zero

Pole order q at a

slide16

Branch Points

Many Value Function

Single

slide19

Cauchy’s Theorem

ถ้ามีฟังก์ชั่นใดที่เป็น Analytic ภายในหรือบน closed contour, integration รอบ contour จะได้ศูนย์

Stake’s theorem

Cauchy – Riemann conditions integral ทางด้านขวามือจะเป็นศูนย์

slide20

ตามเส้นทาง AB หรือ รอบเส้นทาง ACDB

path AB

slide21

curve ACDB

1. ตาม AC

slide22

2. เส้นโค้ง CDB ซึ่งมี constant radius 10

ผลรวมของ Integral

slide23

Example 2 Evaluate

around a circle with its center at the origin.

Although the function is not analytic function

slide24

Example 3 Evaluate

around a circle with its center at the origin.

This result is one of the fundamentals

of contour integration

slide25

Cauchy’s Integral formula

f(a) =constant at g

slide26

The theory of Residue

Pole at origin

Laurent expansion

slide27

Example 1 Evaluate

Around a circle center at the origin

if

Function is analytic

There is a pole order 3 at z = a

if

slide28

Evaluation without Laurent expansion

Many poles : independently evaluate

slide29

Example 2 Evaluate the residues of

Poles at 3,-4

Sum of Residues = 1

slide31

Example 4 evaluate

Around circle and

Pole at z = 0

slide32

Multiple Poles

Dividing throughout by