Class notes EMA51001 (Marhs)

DEPARTMENT OF MATHEMATICS
EMA51002- ANALYTICAL MATHEMATICS
Module 3 – Complex Integration
PART –A
BTL
1. State Cauchy’s Integral Theorem. 1
2. Evaluate  e 2z z = 3 using Cauchy’s integral formula. 1
dz where C is
C  z + 2
4


3. Evaluate ∫
sin 𝜋𝑧
𝑑𝑧 if C is | z | = 2. 2
𝑧−1
𝐶
1 2
4. Expand f ( z )  at z = 2 as a Taylor’s series.
z2
z 2
5. Find the poles of f(z) = 2
.
z - 3z + 2
z 2
6. Find the poles and residues of f(z) = 2
.
z - 3z + 2
𝒛
7. Find the poles of 𝒇(𝒛) = (𝒛−𝟏)(𝒛−𝟑). 1
8. State Cauchy’s Residue theorem. 1
4 2
9. Find the value of  (z - 4)3 (z - 2) dz , where C is | z | = 3.
C
10. Define singular point. 1


PART B
𝒛𝟐 3
1. Evaluate :∫𝑪 (𝒁−𝟏)𝟐 (𝒁−𝟐)
𝑑𝑧 where C is |𝑧| = 3, using Cauchy’s
integral formula.
𝒛𝟐 +𝟏 3 3
2. Evaluate ∫𝑪 (𝒁−𝟏)(𝒁−𝟐)
𝑑𝑧 where C is |𝑧| = , using Cauchy’s
2
integral formula.