LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
SIXTH SEMESTER – APRIL 2012
MT 6603/MT 6600 - COMPLEX ANALYSIS
Date : 16-04-2012
Dept. No.
Max. : 100 Marks
Time : 1:00 - 4:00
PART - A
Answer all questions:
1. Find the absolute value of
(10× 𝟐 = 𝟐𝟎)
(1+3𝑖)(1−2𝑖 )
3+4𝑖
2. Define harmonic function.
3. Find the radius of convergence of the series ∑∞
𝑛=1
4. Using Cauchy integral formulas evaluate∫𝐶
𝑒𝑧
𝑧
𝑧𝑛
𝑛
.
where 𝐶 is the unit circle |𝑧| = 1.
5. Define zero and poles of a function.
6. Write Maclaurin series expansion of s𝑖𝑛 𝑧.
7. Define residue of a function.
8. State Argument principle.
9. Define isogonal mapping.
10. Define critical point.
PART - B
Answer any FIVE questions:
𝑥𝑦
11. Let f(z)= {
𝑥 2 −𝑦 2
(5× 𝟖 = 𝟒𝟎)
if 𝑧 ≠ 0
. Show that f(z) satisfies CR equations at zero but not
0
if 𝑧 = 0
differential at 𝑧 = 0.
12. Prove that 𝑢 = 2𝑥 − 𝑥 3 + 3𝑥𝑦 2 is harmonic and find its Harmonic conjugate.
13. State and prove Liouvilles theorem and deduce Fundamental theorem of algebra.
14. Find the Taylors series to represent
𝑧 2 −1
in
(𝑧+2)(𝑧+3)
|𝑧| < 2.
15. Suppose 𝑓(𝑧) is analytical in the region 𝐷 and is not identically zero in 𝐷.Show that the
set of all zeros of 𝑓(𝑧) is isolated.
3cos z
dz dz over 𝐶 where 𝐶 is the unit circle.
16. Use residue calculus to evaluate 
2i  3z
C
17. Show that any bilinear transformation can be expressed as a product of translation,
rotation, magnification or contraction and inversion.
18. Find the bilinear transformation which maps the points 𝑧1 = 2, 𝑧2 = 𝑖, 𝑧3 = − 2 onto
𝑤1 = 1 , 𝑤2 = 𝑖, 𝑤3 = −1.
PART – C
Answer any TWO questions:
(2 × 𝟐𝟎 = 𝟒𝟎)
19. (a) Derive CR equations in polar coordinates.
(b) Find the real part of the analytic function whose imaginary part is𝑒 −𝑥 (2 xy cos y +
(y 2 − x 2 )sin y)and construct the analytic function.
20 .
(a) State and prove Cauchy integral formula.
(b) Let F be an analytic inside and on a simple closed curve C. Let z be a point inside C.
Show that f’(z) =∫𝐶
𝑓(𝑡)
(𝑧−𝑡)2
dt.
𝑧
21. (a) Expand𝑓(𝑧) = (𝑧−1)(2−𝑧) in a Laurants series in (i) 1 < |𝑧| < 2 , ( ii) |𝑧 − 1| > 1 .
∞
𝑥2
(b) Using method of contour integration evaluate ∫−∞ (𝑥 2 +1 ) (𝑥 2 + 4) dx.
22. (a) Show that any bilinear transformation which maps the unit circle |𝑧| = 1 onto |𝑤| = 1
𝑧− 𝛼
can be written in the form 𝑤 = 𝑒 𝑖𝜆 (𝛼̅𝑧−1) where 𝜆is real.
(b)State and prove Rouche’s theorem.
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