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(1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of
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1. Section 5 SECTION 5 Complex Integration II (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function

2. Section 5 value of the integral between two points depends on the path no real meaning to

3. Example Section 5 integrate the function along the path C joining 2 to 12jas shown

4. Example Section 5 integrate the function along the path CC1 C2 joining 2 to 12jas shown Along C1: along real axis ! Along C2:

5. Section 5 value of the integral along both paths is the same coincidence ??

6. Dependence of Path Section 5 Suppose f (z) is analytic in a simply connected domain D by the Cauchy Integral Theorem note: if they intersect, we just do this to each “loop”, one at a time

7. Integration (independence of path) Section 5 Consider the integral If f (z) is analytic in a simply connected domain D, and z0 and z1 are in D, then the integral is independent of path in D Not only that, but....... where e.g.

8. Examples Section 5 the whole complex plane (1) (2) ( f (z) not analytic anywhere - dependent on path ) (3) f (z) analytic in this domain (both 1z2and 1z are not analytic at z0 - the path of integration C must bypass this point)

9. Section 5 Question: Can you evaluate the definite integral

10. More Integration around Closed Contours ... Section 5 We can use Cauchy’s Integral Theorem to integrate around closed contours functions which are (a) analytic, or (b) analytic in certain regions For example, f (z) is analytic everywhere except at z0 But what if the contour surrounds a singular point ?

11. Cauchy’s Integral Formula Section 5 Let f (z) be analytic in a simply connected domain D. Then for any pointz0 in D and any closed contour C in D that encloses z0

12. Cauchy’s Integral Formula Section 5 Let f (z) be analytic in a simply connected domain D. Then for any pointz0 in D and any closed contour C in D that encloses z0

13. Example Section 5 Evaluate the integral where C is Singular point inside ! The Cauchy Integral formula becomes or

14. Example Section 5 Evaluate the integral where C is Singular point inside ! The Cauchy Integral formula becomes or

15. Example Section 5 Evaluate the integral where C is Singular point inside ! The Cauchy Integral formula becomes or

16. Example Section 5 Evaluate the integral where C is Singular point inside ! The Cauchy Integral formula becomes or

17. Illustration of Cauchy’s Integral Formula Section 5 Let us illustrate Cauchy’s Integral formula for the case of f (z)z and z0 1 So the Cauchy Integral formula becomes f (z) is analytic everywhere, so C can be any contour in the complex plane surrounding the point z1 or

18. Another Example Section 5 where C is any closed contour surrounding zj Evaluate The Cauchy Integral formula becomes or f (z) is analytic everywhere

19. Another Example Section 5 where C is any closed contour surrounding zj Evaluate The Cauchy Integral formula becomes or f (z) is analytic everywhere

20. Another Example Section 5 Let us illustrate Cauchy’s Integral formula for the case of f (z)1 and z0 0 So the Cauchy Integral formula becomes f (z) is a constant, and so is entire, so C can be any contour in the complex plane surrounding the origin z0 or

21. Another Example Section 5 Let us illustrate Cauchy’s Integral formula for the case of f (z)1 and z0 0 So the Cauchy Integral formula becomes f (z) is a constant, and so is entire, so C can be any contour in the complex plane surrounding the origin z0 or

22. Let us now prove Cauchy’s Integral formula for this same case: f (z)1 and z0 0 Section 5 Cut out the point z0 from the simply connected domain by introducing a small circle of radius r - this creates a doubly connected domain in which 1z is everywhere analytic. From the Cauchy Integral Theorem as applied to Doubly Connected Domains, we have But the second integral, around C*, is given by note: see section 4, slide 6

23. Equations involving the modulus Section 5 (these are used so that we can describe paths (circles) of integration more concisely) What does the equation mean ? mathematically: equation of a circle

24. Example Section 5 equation of a circle

25. Section 5

26. Section 5

27. Section 5 centre

29. Question: Section 5

30. Examples Section 5 Evaluate the following integrals: (1) where C is the circle z 2 let let f (z) is analytic in D and C encloses z0

31. Section 5 (2) where C is the circle zj1 First of all, note that 1(z21) has singular points at zj. The path C encloses one of these points, zj. We make this our point z0 in the formula We need a term in the form 1(z z0) so we rewrite the integral as:

32. Section 5 let

33. Section 5 let let

34. Section 5 let let

35. Section 5 (3) where C is the circle zj1 Here we have The path C encloses one of the four singular points, zj. We make this our point z0 in the formula where Now

36. Section 5 Question: Evaluate the integral where C is the circle z 2 (i) Where is C ? (ii) where are the singular point(s) ? (ii) what’s z0 and what’s f (z) ? Is f (z) analytic on and inside C ? (iii) Use the Cauchy Integral Formula.........

37. Section 5 (4) where C is the circle z3/2 tanz is not analytic at /2, 3/2, , but these points all lie outside the contour of integration The path C encloses two singular points, z1. To be able to use Cauchy’s Integral Formula we must only have one singular point z0inside C. Use Partial Fractions:

38. Section 5

39. Generalisation of Cauchy’s Integral Formula Section 5 More complicated functions, having powers of z-z0, can be treated using the following formula: f analytic on and inside C, z0 inside C For example, This formula is also called the “formula for the derivatives of an analytic function” Note: when n0 we have Cauchy’s Integral Formula:

40. Example Section 5 Evaluate the integral where C is the circle z 2 let let f (z) is analytic in D, and C encloses z0

41. Example Section 5 Evaluate the integral where C is the circle z 2 let let f (z) is analytic in D, and C encloses z0

42. Example Section 5 Evaluate the integral where C is the circle z 2 let let f (z) is analytic in D, and C encloses z0

43. Another Example Section 5 Evaluate the integral where C is the circle z 2 let let f (z) is analytic in D, and C encloses z0

44. Summary of what we can Integrate Section 5 (1) with f (z) analytic inside and on C - equals 0 ( Cauchy’s Integral Theorem ) (2) with f (z) analytic inside and on C, except at zzo - equals ( Cauchy’s Integral Formula ) (3) with f (z) analytic inside and on C, except at zzo ( The Formula for Derivatives )

45. Summary of what we can Integrate Section 5 (1) with f (z) analytic inside and on C - equals 0 ( Cauchy’s Integral Theorem ) (2) with f (z) analytic inside and on C, except at zzo - equals ( Cauchy’s Integral Formula ) (3) with f (z) analytic inside and on C, except at zzo ( The Formula for Derivatives )

46. Summary of what we can Integrate Section 5 (1) with f (z) analytic inside and on C - equals 0 ( Cauchy’s Integral Theorem ) (2) with f (z) analytic inside and on C, except at zzo - equals ( Cauchy’s Integral Formula ) (3) with f (z) analytic inside and on C, except at zzo ( The Formula for Derivatives )

47. Summary of what we can Integrate Section 5 (1) with f (z) analytic inside and on C - equals 0 ( Cauchy’s Integral Theorem ) (2) with f (z) analytic inside and on C, except at zzo - equals ( Cauchy’s Integral Formula ) (3) with f (z) analytic inside and on C, except at zzo ( The Formula for Derivatives )

48. Section 5 What can’t we Integrate ? Functions we can’t put in the form of our formulas: e.g. where C is (singularities at 2 inside C) e.g. where C is the unit circle (singularity at 0 inside C)

49. Topics not Covered Section 5 (1) Proof that the antiderivative of an analytic function exists where (use the MLinequality in the proof) (2) Proof of Cauchy’s Integral Formula (use the MLinequality in the proof) (3) Proof that the derivatives of all orders of an analytic function exist - and the derivation of the formulas for these derivatives (use Cauchy’s Integral Formula and the MLinequality in the proof)

50. (4) Morera’s Theorem (a “converse” of Cauchy’s Integral Theorem) Section 5 “If f (z) is continuous in a simply connected domain D and if for every closed path in D, then f (z) is analytic in D” (5) Cauchy’s Inequality (proved using the formula for the derivatives of an analytic function and the MLinequality) (6) Liouville’s Theorem “If an entire function f (z) is bounded in absolute value for all z, then f (z) must be a constant” - proved using Cauchy’s inequality