SECTION 8 Residue Theory (1) The Residue (2) Evaluating Integrals using the Residue (3) Formula for the Residue (4) The Residue Theorem 1. We need to be able to take the derivative in complex analysis. We wouldn’t be able to do much otherwise. E.g. if we have displacement we can use calculus to obtain s and a, which are really useful/essential to know. 2. But the derivative in complex analysis is a bit trickier than the derivative in “ordinary” real calculus

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:

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:

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) C So We can use it to evaluate integrals

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) C So We can use it to evaluate integrals

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

Section 8 singular point centre

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

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

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

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

So the Residue allows us to evaluate integrals of analytic 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

That’s great - but every time we want to evaluate an integral 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.

Formula for finding the residue for a simple pole 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

Formula for finding the residue for a simple pole 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

Formula for finding the residue for a simple pole 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

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

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

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

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

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

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

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

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:

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

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:

The Residue Theorem 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 C 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 C

Section 8 Example Integrate the function around C

Section 8 Example Integrate the function around C

Section 8 Example Integrate the function around C

Section 8 Example Integrate the function around C

Section 8 Example Integrate the function around C

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

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 1. We need to be able to take the derivative in complex analysis. We wouldn’t be able to do much otherwise. E.g. if we have displacement we can use calculus to obtain s and a, which are really useful/essential to know. 2. But the derivative in complex analysis is a bit trickier than the derivative in “ordinary” real calculus e.g.