Reg No.: Name:

Max. Marks: 100

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A3801 Pages: 2

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
THIRD SEMESTER B.TECH DEGREE EXAMINATION, APRIL 2018
Course Code: MA201
Course Name: LINEAR ALGEBRA AND COMPLEX ANALYSIS

PART A
Answer any two full questions, each carries 15 marks
Let f(z) = u(x, 9) + iv(x, 9) be defined and continuous in some neighbourhood
ofa point 2 = 2 + 70 and differentiable at 2 itself. Then prove that the first
order partial derivatives of wand v exist and satisfy the Cauchy — Riemann
equations.
Prove that ४ = sin x cosh y is harmonic. Hence find its harmonic conjugate.

7 : ⋅ ↥ 1 ⋅ 1
Find the image of the region |2 − न ಡ್ಯ under the transformation w = ட

Find a linear fractional transformation which maps —1,0,1 onto 1,1+ i,1+ 27.
Re (25)

⋅ −−≀∎∅≠∘⋅ ⋅

∁↥↿∁∁⇂⊓∨↥↧⊜↥∣↧⊜∩↥↧⊜∊⋯↴∁↕↥∘∐∫≼∑⋟−−∣∑∣⇄ f is continuous at z = 0.
0 ifz=0

Find the image of the x-axis under the linear fractional transformation w = a
PART B

Answer any two full questions, each carries 15 marks

Evaluate ‏مل‎ Im(z*)dz where C is the triangle with vertices 0,1, counter-

clockwise.

Using Cauchy’s Integral Formula, evaluate J. 7 dzwhere c is taken

25-23-24.
counter-clockwise around the circle:

ட 1411-2 ii) |2-1-1 =>
Determine and classify the singular points for the following functions:
⋅ 2 ட 5112 ‏لاد‎ = (ടു)
1) /(2) = Gam ii) g(z) = (z+ i)*e\#
E o 41
valuate [ചിന

E tan 2 ⋅ 3 ⋅ ⋅
valuate f ल्क्राए dz counter clockwise around ८: |z| = > using Cauchy’s Residue

Theorem.
-2243

5 with centre 0 in
254-3212

Find all Taylor series and Laurent series of f(z) =
i) |zl< 1 ii) 1< |2| <2.

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Duration: 3 Hours

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