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2 C 20646

State Rouche’s theorem.
sinz
Find the residue of a atz=0.

How many zeroes of are in the disc |z| = 1 for the function f(z) = 29 - 822 +5.
(10 x 3 = 30 marks)
Section B

Answer at least five questions.

Each question carries 6 marks.

All questions can be attended.
Overall Ceiling 30.

Check whether the function U is harmonic or not if so find its harmonic conjugate
U (a, y) =x? — Bay? — By.

Find all the solutions of the equation sin z = 5.
State and prove Fundamental theorem of algebra.
State and prove Morera’s theorem.

. . : . 1
Find the Taylor's series expansion with centre 2 = 2i of f(z) = Tos

1
ify t . int ify t sin Ae)
Identify the singular points and classify them f(z) = ப. அஜ
2(1+2

Find residue of oe atz=0.

०0 1
oF —— dx.
Find | +1

(5 x 6 = 30 marks)
Section C (Essay Questions)

Answer any two questions.
Each question carries 10 marks.

State and prove Cauchy Riemann Equation. Also state the sufficient condition for differentiability.

State and prove Cauchy's integral formula for derivatives.

1
Expand f(z)= 2(2-1) in a Laurent series valid for 1 < |2-2| < 2.

State and prove Cauchy's residue theorem.
(2 x 10 = 20 marks)

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