Test 3: Complex Analysis
Instruction: Create a pdf file with all the solutions, upload file on Google Drive and share the link in the form shared at the bottom of the page
Total Marks: 75
1) Define the following terms with an example of each:
i) Analytic Functions
ii) Harmonic Functions
1.(iii) State CR equations and explain how these are helpful in explaining if a given function is analytic or not.
1.(iv) Explain Milne Thomson method.
1.(v) State Rouche’s Theorem and give an example of how it can be applied.
[10M]
2) Define Singularity and explain below types of singularities with an example of each: i) Isolated Singularity ii) Removable Singularity iii) Essential Singularity iv) Pole
[10M]
3.(i) In the context of Complex Analysis, write 1 sentence including the below statement “necessary but not sufficient”.
3.(ii) Define simply connected and multiply connected domains.
[4M]
4) State and prove Cauchy’s Integral theorem.
[10M]
5) State and prove Cauchy’s Integral formula. Use it to derive the expression for Cauchy’s integral formula for derivatives.
[10M]
6.(i) Define a power series and explain its radius of convergence.
6.(ii) State the conditions under which a function can be expressed as a Taylor series and write the expression for Taylor’s series.
6.(iii) State the conditions under which a function can be expressed as a Taylor series and write the expression for Laurent’s series.
[9M]
7) Define Residue and write the expression for calculating residues in each of the following cases:
(i) at simple pole (ii) at pole of order \(m\) (iii) at infinity
[8M]
8.(a) State and prove Cauchy’s Residue Theorem.
8.(b) With regards to Contour Integration, (i) State Cauchy’s Lemma (ii) State Jordan’s Lemma
[14M]