Math Department complex analysis 413 Science collage
King Abdul Aziz University Dr. Najwa Joharji
Objectives of chapter 4 (Integrals)
1. Evaluate the integral of f(z) using the theorem (Indefinite Integration of analytic function). ( ) ( 1) ( 0), ( ) ( ) .
1
0
z f z F where z
F z F dz z f
z
z
2. Find parametric equations of some curves.
3. Evaluate Line Integral using a parametric equation of the path.
4. Verify dependence on the path.
5. Show that f z dz ML whenever f z M
C
( ) , ( ) .6. Define a simple closed path.
7. Define a simply connected domain.
8. Use the C.I.Theorem to show that when the function f(z0 is analytic in C.
C
dz z
f( ) 0
9. Define independence of the path.
10. Use the C.I.Formula to evaluate the integral 2 ( ) )
( ) (
0 0
z if z dz
z z f
C
.11. Use the partial fraction together with the C.I.Formula to evaluate the integral of a function has more than one singularity inside C.
12. Use The Derivative Formula to evaluate the integral when the function has singularity of order grate than
one.
z (z0)
1dz 2n!i f( )(z0)z
f n
C
n
.1
Math Department complex analysis 413 Science collage
King Abdul Aziz University Dr. Najwa Joharji
Objectives of Chapter 5 (Series)
1. Define a power series.
2. Find the interval of convergence of a power series using Ratio test 3. Define Taylor series for an analytic function.
4. Find the Taylor series and Maclaurin series for some analytic functions.
5. Expand an analytic function into a Taylor series and Maclaurin series 6. Use Taylor series and Maclaurin series to evaluate limits.
7. Deduce Taylor series and Maclaurin series for some functions by differentiation or integration of famous ones.
8. Define the Laurent series for a complex function.
9. Find the Laurent series for some functions and where they valid.
10. Find the principal part of a Laurent series.
11. Find the residue of a function at a point using Laurent series.
12. Classify singularities using the principal part of Laurent series.
13. Define the three types of singularities.
2
Math Department complex analysis 413 Science collage
King Abdul Aziz University Dr. Najwa Joharji
3 Objectives of chapter 6(Residues and Poles)
1. Use the Residue Theorem to evaluate
.C
dz z f( )
2. Find the residue when z0 is a simple pole of f(z)by two methods 1.
Re
( )lim
( 0) ( ).0 0
z f z z z
f
z z z
z
s
2. When
) (
) ) (
( Q z
z z P
f ,P(z) Q(z) are analytic functions
) (
) ) (
( ,
0 ) ( 0 ) (
,
Re lim
0 0
0
0 Q z
z z P
f then
z andQ z
P
z z z
z
s
. 3. Find the residue of f(z) has a pole of order m
) ( ) (
0 )!
1 ( ) 1
(
lim [
0Re
0 0
z f z m z
m z
m z z
f m
z
z
dz
s d
.