R3902

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Reg ٥٥: ‏سس‎ Name:"

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
THIRD SEMESTER B.TECH DEGREE EXAMINATION, DECEMBER 201 8

Course Code: MA201 course Name: LINEAR ALGEBRA AND

COMPLEX ANALYSIS

Max. Marks: 100 Duration: 3 Hours

PARTA
Answer any twofull questions, each carries 15 marks Marks

1 a) Prove that f(z) =e Y+iY is analytic. Find ۶ (2). (7) b) Show that ए = 3x ‏پ2‎ — * is harmonic. Also
find the harmonic conjugate ofv. (8) 2 a) Find the linear fractional transformation that maps 71 = O,
22 = 1173 = 00 (8) onto WI = —l,wz = —t,W3 = 1 respectively.

b) Find the image of the lines x = a and y = b where a and b are constants, under (7) the
transformation w = 22

3 a) If f(Cz) —u + iv is analytic, prove that प = €, and v = ‏2ء‎ are families of curves (7) cutting

orthogonally.
b) Prove that w = -22 maps the upper half plane (५०0) into the
interior of IWI=1 (8)
PART 8
Answer any two full questions, each carries 15 marks
4 93) Expand f(z) = — as Taylor's series about 2 = 2 (7)
b) Evaluate c cos (z-D(z-2)*Rin dz where C is 171 = 3, using
Cauchy's integral formula. (8)
3-23 Pe
5 9) Evaluate $= z—43—5 where ¢; 12 — 2 — ।| = 3.2, using Cauchy's residue (7)
theorem.
co 1 33
f dx =—
b) Show that ~~ © { കയി? 8 (8)
6 a)(7) 5
Find the Laurent's series expansion of f (Z) = Gai aboutz about 2 = -|
0) Find the poles ಹ್‌ and residues of the function
f(z) (8) - ಜಾರ್‌

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