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

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

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

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**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 integrate the function along the path C joining 2 to 12jas shown**Example**Section 5 integrate the function along the path CC1 C2 joining 2 to 12jas shown Along C1: along real axis ! Along C2:**Section 5**value of the integral along both paths is the same coincidence ??**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**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.**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 1z2and 1z are not analytic at z0 - the path of integration C must bypass this point)**Section 5**Question: Can you evaluate the definite integral**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 z0 But what if the contour surrounds a singular point ?**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**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**Example**Section 5 Evaluate the integral where C is Singular point inside ! The Cauchy Integral formula becomes or**Example**Section 5 Evaluate the integral where C is Singular point inside ! The Cauchy Integral formula becomes or**Example**Section 5 Evaluate the integral where C is Singular point inside ! The Cauchy Integral formula becomes or**Example**Section 5 Evaluate the integral where C is Singular point inside ! The Cauchy Integral formula becomes or**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 z1 or**Another Example**Section 5 where C is any closed contour surrounding zj Evaluate The Cauchy Integral formula becomes or f (z) is analytic everywhere**Another Example**Section 5 where C is any closed contour surrounding zj Evaluate The Cauchy Integral formula becomes or f (z) is analytic everywhere**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 z0 or**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 z0 or**Let us now prove Cauchy’s Integral formula**for this same case: f (z)1 and z0 0 Section 5 Cut out the point z0 from the simply connected domain by introducing a small circle of radius r - this creates a doubly connected domain in which 1z 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**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**Example**Section 5 equation of a circle**Section 5**centre**Section 5**radius centre**Question:**Section 5**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**Section 5**(2) where C is the circle zj1 First of all, note that 1(z21) has singular points at zj. The path C encloses one of these points, zj. 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:**Section 5**let**Section 5**let let**Section 5**let let**Section 5**(3) where C is the circle zj1 Here we have The path C encloses one of the four singular points, zj. We make this our point z0 in the formula where Now**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.........**Section 5**(4) where C is the circle z3/2 tanz is not analytic at /2, 3/2, , but these points all lie outside the contour of integration The path C encloses two singular points, z1. To be able to use Cauchy’s Integral Formula we must only have one singular point z0inside C. Use Partial Fractions:**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 n0 we have Cauchy’s Integral Formula:**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**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**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**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**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 zzo - equals ( Cauchy’s Integral Formula ) (3) with f (z) analytic inside and on C, except at zzo ( The Formula for Derivatives )**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 zzo - equals ( Cauchy’s Integral Formula ) (3) with f (z) analytic inside and on C, except at zzo ( The Formula for Derivatives )**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 zzo - equals ( Cauchy’s Integral Formula ) (3) with f (z) analytic inside and on C, except at zzo ( The Formula for Derivatives )**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 zzo - equals ( Cauchy’s Integral Formula ) (3) with f (z) analytic inside and on C, except at zzo ( The Formula for Derivatives )**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)**Topics not Covered**Section 5 (1) Proof that the antiderivative of an analytic function exists where (use the MLinequality in the proof) (2) Proof of Cauchy’s Integral Formula (use the MLinequality 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 MLinequality in the proof)**(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 MLinequality) (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