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# Chap 4 Complex Algebra. For application to Laplace Transform Complex Number.

## งานนำเสนอเรื่อง: "Chap 4 Complex Algebra. For application to Laplace Transform Complex Number."— ใบสำเนางานนำเสนอ:

Chap 4 Complex Algebra

For application to Laplace Transform Complex Number

Argand Diagram r  x y

Complex Variables Continuous Function Cplxdemo.m

Single Value Function Many Values Function

Derivatives of Complex Variables 1 0

10 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 z 0, satisfies the Cauchy Riemann Conditions

Example

Cauchy Riemann Conditions

At Origin Keep y constant 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

Essential Singularities E_1_z.m

Branch Points Many Value Function Single

4.13 INTEGRATION OF FUNCTION OF COMPLEX VARIABLES

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 

The theory of Residue Pole at origin Laurent expansion

Example 1 Evaluate if Around a circle center at the origin Function is analytic if There is a pole order 3 at z = a

Evaluation without Laurent expansion Many poles : independently evaluate

Example 2 Evaluate the residues of Poles at 3,-4 Sum of Residues = 1

If the denominator does not factorize L’Hopital’s rule

Example 4 evaluate Around circle and Pole at z = 0

Multiple Poles Dividing throughout by

Example 5 Evaluate

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