A B3A001S Pages: 2

Reg. No. Name:
APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
THIRD SEMESTER B.TECH DEGREE EXAMINATION, JULY 2017

Course Code: MA 201
Course Name: LINEAR ALGEBRA AND COMPLEX ANALYSIS.

Max. Marks :100 Duration: 3 hours
PART A

Answer any two questions.

1. (a) Does the limit 11೫, = exit? If yes find the value. If no, explain why? (8)
(0) If f(z) = ॥ + iv is analytic, prove that 1८ = constant and v = constant are families of
curves cutting orthogonally (7)
2. (a) Find the image of the semi-circle y = +v4 — x? under the transformation w = 2
(7)
(b) Find the image of the half-plane Re(z) > 2 under the map w = iz (8)
3. (a) Find the points, if any, in complex plane where the function f(z) = 2x? +y +
i(y? — x) is
(i) differentiable (ii) analytic. (8)
(b) Prove that the function u(x,y) = x? - 3xy? - 59 is harmonic everywhere. Also
find the harmonic conjugate of u. (7)
PART B
Answer any two questions.
4. (a) Evaluate ‏مل‎ 202 where 0 is given by = 3t,y=t?,-1
(b) Show that f ८ 2 + 2)2ദ്മ = ಪ where C is any path connecting the points -2 and
2+i (7)

5247 ⋅ ⋅
5. (a) Evaluate J. ராணா dz where C is the circle |z — 2| = 2. (8)
(b) Find the Laurent’s series expansion of 111 <|z+1|<2. (7)
೧ ५ 2+1 ⋅ 5

6. (a) Use Cauchy’s integral formula to evaluate J. ಸರ್‌ dz where C is |2| = 1.

(8)
⋅ ⋅ ⋅ ∞∝⋮−≈⊸⊦∑

(b) Using Contour integration, evaluate क नर तल ന്‌ (7)

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