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

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

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

4 integrate the function along the path C  C 1  C 2 joining 2 to 1  2j as shown Example Along C 1 : along real axis ! Along C 2 : Section 5

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

6 Dependence of Path 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 Section 5

7 Integration (independence of path) Consider the integral If f (z) is analytic in a simply connected domain D, and z 0 and z 1 are in D, then the integral is independent of path in D where e.g. Not only that, but

8 Section 5 Examples (1) the whole complex plane (2) ( f (z) not analytic anywhere - dependent on path ) (3) f (z) analytic in this domain (both 1  z 2 and 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 Section 5 More Integration around Closed Contours... 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 Section 5 Cauchy’s Integral Formula Let f (z) be analytic in a simply connected domain D. Then for any point z 0 in D and any closed contour C in D that encloses z 0

12 Section 5 Cauchy’s Integral Formula Let f (z) be analytic in a simply connected domain D. Then for any point z 0 in D and any closed contour C in D that encloses z 0

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

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

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

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

17 Section 5 Illustration of Cauchy’s Integral Formula Let us illustrate Cauchy’s Integral formula for the case of f (z)  z and z 0  1 So the Cauchy Integral formula becomes or f (z) is analytic everywhere, so C can be any contour in the complex plane surrounding the point z  1

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

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

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

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

22 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 note: see section 4, slide 6 Let us now prove Cauchy’s Integral formula for this same case: f (z)  1 and z 0  0 But the second integral, around C *, is given by

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

24 Section 5 Example equation of a circle

25 Section 5

26 Section 5

27 Section 5 centre

28 Section 5 centre radius

29 Section 5 Question:

30 Section 5 Examples Evaluate the following integrals: (1) where C is the circle  z  2 let f (z) is analytic in D and C encloses z 0

31 Section 5 (2) where C is the circle  z  j  1 We need a term in the form 1  (z  z 0 ) so we rewrite the integral as: First of all, note that 1  (z 2  1) has singular points at z    j. The path C encloses one of these points, z  j. We make this our point z 0 in the formula

32 Section 5 let

33 Section 5 let

34 Section 5 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 z 0 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 z 0 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 z 0 inside C. Use Partial Fractions:

38 Section 5

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

40 Section 5 Example Evaluate the integral where C is the circle  z  2 let f (z) is analytic in D, and C encloses z 0

41 Section 5 Example Evaluate the integral where C is the circle  z  2 let f (z) is analytic in D, and C encloses z 0

42 Section 5 Example Evaluate the integral where C is the circle  z  2 let f (z) is analytic in D, and C encloses z 0

43 Section 5 Another Example Evaluate the integral where C is the circle  z  2 let f (z) is analytic in D, and C encloses z 0

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

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

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

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

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

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

50 Section 5 (4) Morera’s Theorem (a “converse” of Cauchy’s Integral Theorem) (5) Cauchy’s Inequality “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” (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