1. 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

2. 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:

3. 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:

4. 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

5. 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

6. 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:

7. Section 8
singular
point
centre

8. 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:

9. 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

10. 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:

11. 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:

12. 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

13. 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.

14. 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

15. 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

16. 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

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

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

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

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

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

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

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

24. 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:

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

26. 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:

27. 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

28. Section 8
Example
Integrate the function around C

29. Section 8
Example
Integrate the function around C

30. Section 8
Example
Integrate the function around C

31. Section 8
Example
Integrate the function around C

32. Section 8
Example
Integrate the function around C

33. 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

34. 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.