A 034005 Pages:2

Reg. No. Name:
APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
THIRD SEMESTER B.TECH DEGREE EXAMINATION, MARCH 2017
MA 201: LINEAR ALGEBRA AND COMPLEX ANALYSIS

Max. Marks: 100 Duration: 3 Hours
PART A
Answer any 2 questions
1. a. Check whether the following functions are analytic or not. Justify your answer.
i) /)2(- 2+2 (4)
iy 7 ()-1 (4)
b. Show that 7(2)- sin z is analytic for all 2. Find f’(z) (7)
2. a. Show that v=3x*y—y° is harmonic and find the corresponding analytic function
s(c)=u(x,y)+iv(x,y) 5
1 :

b. Find the image of 0 > × >1 2 > ‏مر‎ <] under the mapping w = €: (7)

3. a. Find the linear fractional transformation that carries 2, = —2, 22 = 0 and 23 = 2

on to the points ۷ = ००, w2 = 1, and w3 = 3/6 Hence find the image of x-axis.(7)

b. Find the image of the rectangular region— छ
w=sinz (8)
PART B
Answer any 2 questions
4, ೩. Evaluate 1 12162 where
i) C is the line segment joining -i and i (3)
ii) © 15 the unit circle in the left of half plane (4)
b. Verify Cauchy’s integral theorem for z* taken over the boundary of the rectangle
with vertices -1, 1, 1+1, - 1+7 in the counter clockwise sense. (8)
5. a. Find the Laurent’s series expansion of f(z)= = which is convergent in
1) |2- 1| <2 (4)
ii) |2- 1| > 2 (4)

b. Determine the nature and type of singularities of

ட > (3)

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