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A7046

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APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
THIRD SEMESTER B.TECH DEGREE EXAMINATION, DECEMBER 2017

Course Code: MA201
Course Name: LINEAR ALGEBRA AND COMPLEX ANALYSIS

PARTA
Answer any two full questions, each carries 15 marks.

Find the points where Cauchy-Riemann equations are satisfied for the function
f(z) = 5५४ + i x? 9. Where does f (7) exist? Is the function f(z) analytic at those
points?

If v=e*(x sin y + $ cos $), find an analytic function f(z)=u+iv.

Show that u = x?-y?-y is harmonic. Also find the corresponding conjugate harmonic
function.

(i) Find a bilinear transformation which maps (—i, 0, i) onto (0, -1, ೦೦).

17722

(ii) Test the continuity at 2 = 0, if f(z) = “a #0

=0,z=0
Find the image of the lines x=1, y=2 and x>0, ൮90 under the mapping W= 22
Find the image of the semi-infinite strip x > 0, 0< y < 2 under the transformation

w=iz+1. Draw the regions.

PART B
Answer any two full questions, each carries 15 marks.

Evaluate டூ Re 2202 over the boundary C of the square with vertices 0, i, 1+ i,1

clockwise
35 4-2 : 3
valuate നന്ന dz over the circle |=
2
Evaluate | പ്‌ 02 over the circle |z+ i |=1
2 ⋅
Expand ಗಾ (1) 002-241, (2) |z-1|>1
‏ا‎
‎Evaluate ‏ہے و‎

Using Residue theorem evaluate | dz over the circle |z|=3

22
(2-1)2(242)

Find the Taylor series of = about the point z= 7

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Duration: 3 Hours

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