Complex Analysis-Ii

Course: M.Sc Code: DU-209 Branch: Mathematics Year: 2021

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DU-209

M.Sc. IInd Semester (CBCS) Examination, 2021

Mathematics

Paper - (204)

Complex Analysis-II

[Maximum Marks: 60]

Note :- All questions from each section carry equal marks. All questions are compulsory and answer limit are approximately 250 words. Start the answer of each section from new page. Maximum limit of pages of answer booklet are approximately 16 pages. Answer should be written by the student in his/her own handwriting mandatory. The first page of answersheet should be download by the student from university website is mandatory.

1.
If $|z|\leq1$ and $p\geq 0$ then prove that \( |1-E_p(z)| \leq |z|^{p+1} \) where $E_p(z)$ is elementary factor.
2.
Find the analytic continuation of the function $f_1$ defined by the series
\( f_1(z) = \sum_{n=0}^\infty z^n \)
3.
Define Harmonic functions and write their basic properites.
4.
Find the order of the function $f(z) = \cos z$.
5.
Define the following :-
  1. Univalent function
  2. Bloch's constant