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(1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of

Section 5. SECTION 5 Complex Integration II. (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function. Section 5. value of the integral between two points depends on the path. no real meaning to. Example. Section 5.

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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

  28. Section 5 radius 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

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