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Chap 4

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

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  1. Chap 4 Complex Algebra

  2. For application to Laplace Transform • Complex Number

  3. Argand Diagram y r q x

  4. Complex Variables Continuous Function Cplxdemo.m

  5. Single Value Function Many Values Function

  6. Derivatives of Complex Variables 1 0

  7. 0 1 Cauchy Riemann Conditions

  8. 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

  9. Example

  10. Example Cauchy Riemann Conditions

  11. Keep y constant At Origin One_OVER_Z.m

  12. Singularities Poles or unessential Essential Branch points

  13. Poles or unessential Singularities Second order Poles Pole at a Pole order p at zero Pole order q at a

  14. Essential Singularities E_1_z.m

  15. Branch Points Many Value Function Single

  16. 4.13 INTEGRATION OF FUNCTION OF COMPLEX VARIABLES

  17. Cauchy’s Theorem ถ้ามีฟังก์ชั่นใดที่เป็น Analytic ภายในหรือบน closed contour, integration รอบ contour จะได้ศูนย์ Stake’s theorem Cauchy – Riemann conditions integral ทางด้านขวามือจะเป็นศูนย์

  18. ตามเส้นทาง AB หรือ รอบเส้นทาง ACDB path AB

  19. curve ACDB 1. ตาม AC

  20. 2. เส้นโค้ง CDB ซึ่งมี constant radius 10 ผลรวมของ Integral

  21. Example 2 Evaluate around a circle with its center at the origin. Although the function is not analytic function

  22. Example 3 Evaluate around a circle with its center at the origin. This result is one of the fundamentals of contour integration

  23. Cauchy’s Integral formula f(a) =constant at g

  24. The theory of Residue Pole at origin Laurent expansion

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

  26. Evaluation without Laurent expansion Many poles : independently evaluate

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

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

  29. Example 4 evaluate Around circle and Pole at z = 0

  30. Multiple Poles Dividing throughout by

  31. Example 5 Evaluate

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