Analytic function : differentiability and analyticity, Cauchy Riemann equations (necessary and sufficient conditions for a function to be analytic), harmonic functions Unit II Complex integration : line integrals, Cauchy integral theorem, Cauchy integral formula, Cauchy integral formula for higher order derivatives. Unit III Singularities and residues : Taylor’s series, Laurent’s series, zeroes and singularities of complex function, residues at the singularities
1) Prove or disprove: The function ?(?) = |?|2 is analytic at origin. Give detailed reason in support of your answer.
2) Let the imaginary part of an analytic function ?(?) is equal to c (a constant). Find ?(?).
3) Evaluate ∫(?2 + ???) ?? from ?(1,1) to ?(2,4) along the curve ? = ?2.
4) Expand cos? about ? = ?/4 .
5) Prove that ? = 2?(? + 1) − 4. Prove that ?(?,?) is harmonic and find ? such that ?(?) = ? + ?? is analytic function.
6) Evaluate ∫(?3 + 2) ??, over the closed curve C, where C: Lower half of the circle |?| = 2.
7) Expand ?(?) = ? ?2−5?+6
in the region 2< |?| <