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(1) The Residue (2) Evaluating Integrals using the Residue (3) Formula for the Residue

Section 8. SECTION 8 Residue Theory. (1) The Residue (2) Evaluating Integrals using the Residue (3) Formula for the Residue (4) The Residue Theorem. What is a Residue?. Section 8. The residue of a function is the coefficient of the term

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(1) The Residue (2) Evaluating Integrals using the Residue (3) Formula for the Residue

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  1. Section 8 SECTION 8 Residue Theory (1) The Residue (2) Evaluating Integrals using the Residue (3) Formula for the Residue (4) The Residue Theorem

  2. What is a Residue? Section 8 The residue of a function is the coefficient of the term in the Laurent series expansion (the coefficient b1). Examples:

  3. What is a Residue? Section 8 The residue of a function is the coefficient of the term in the Laurent series expansion (the coefficient b1). Examples:

  4. C What’s so great about the Residue? Section 8 The formula for the coefficients of the Laurent series says that (for f (z) analytic inside the annulus) So We can use it to evaluate integrals

  5. C What’s so great about the Residue? Section 8 The formula for the coefficients of the Laurent series says that (for f (z) analytic inside the annulus) So We can use it to evaluate integrals

  6. Example (1) Section 8 Integrate the function counterclockwise about z 2 singular point centre By Cauchy’s Integral Formula:

  7. Section 8 singular point centre

  8. Example (1) Section 8 Integrate the function counterclockwise about z 2 singular point centre By Cauchy’s Integral Formula:

  9. Example (1) cont. Section 8 We could just as well let the centre be at z1 - a one-term Laurent series centre / singular point - as before

  10. Example (2) Section 8 Integrate the function counterclockwise about z 3/2 By Cauchy’s Integral Formula: 0

  11. Example (2) Section 8 Integrate the function counterclockwise about z 3/2 By Cauchy’s Integral Formula: 0

  12. Section 8 So the Residue allows us to evaluate integrals of analytic functions f (z) over closed curves C when f (z) has one singular point inside C. C b1 is the residue of f (z) at z0

  13. Section 8 That’s great - but every time we want to evaluate an integral do we have to work out the whole series ? No - in the case of poles - there’s a quick and easy way to find the residue We’ll do 3 things: 1. Formula for finding the residue for a simple pole 2. Formula for finding the residue for a pole of order 2 3. Formula for finding the residue for a pole of any order e.g. e.g. e.g.

  14. Section 8 Formula for finding the residue for a simple pole If f (z) has a simple pole at z0, then the Laurent series is we’re putting the centre at the singular point here

  15. Section 8 Formula for finding the residue for a simple pole If f (z) has a simple pole at z0, then the Laurent series is we’re putting the centre at the singular point here

  16. Section 8 Formula for finding the residue for a simple pole If f (z) has a simple pole at z0, then the Laurent series is we’re putting the centre at the singular point here

  17. Example (1) Section 8 Find the residue of at zj Check: the Laurent series is

  18. Example (2) Section 8 Find the residue at the poles of Check: the Laurent series are

  19. Example (2) Section 8 Find the residue at the poles of Check: the Laurent series are

  20. Example (2) Section 8 Find the residue at the poles of Check: the Laurent series are

  21. Example (2) Section 8 Find the residue at the poles of Check: the Laurent series are

  22. Example (2) Section 8 Find the residue at the poles of Check: the Laurent series are

  23. Question: Section 8 Find the residue at the pole z01 of

  24. Formula for finding the residue for a pole of order 2 Section 8 If f (z) has a pole of order 2 at z0, then the Laurent series is now differentiate:

  25. Example Section 8 Find the residue of at z1 Check: the Laurent series is

  26. Formula for finding the residue for a pole of any order Section 8 If f (z) has a pole of order m at z0, then the Laurent series is now differentiate m1 times and let zz0 to get:

  27. The Residue Theorem C C Section 8 We saw that the integral of an analytic function f (z) over a closed curve C when f (z) has one singular point inside C is b1 is the residue of f (z) at z0 Residue Theorem: Let f (z) be an analytic function inside and on a closed path C except for at k singular points inside C. Then

  28. C Section 8 Example Integrate the function around

  29. C Section 8 Example Integrate the function around

  30. C Section 8 Example Integrate the function around

  31. C Section 8 Example Integrate the function around

  32. C Section 8 Example Integrate the function around

  33. Proof of Residue Theorem Section 8 Enclose all the singular points with little circles C1, C1, Ck. f (z) is analytic in here By Cauchy’s Integral Theorm for multiply connected regions: But the integrals around each of the small circles is just the residue at each singular point inside that circle, and so

  34. Section 8 Topics not Covered (1) Another formula for the residue at a simple pole (when f (z) is a rational function p(z)q(z), (2) Evaluation of real integrals using the Residue theorem e.g. using (3) Evaluation of improper integrals using the Residue theorem e.g.

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