United Arab Emirates University
College of Sciences
Department of Mathematical Sciences
FINAL EXAM
Complex Analysis I
MATH 315 SECTION
01
CRN 23516
9:30 { 10:45 on Monday & Wednesday
Date: Wednesday, January 6, 2010
ID No:
200
Name:
Score:
/70
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These are the notes of Exam of Complex Analysis and its key important points are: Imaginary Parts, Express, Form, Real and Imaginary Parts, Principal Branch, Singular Points, Region, Image, Mapping, Function
Typology: Exams
2012/2013
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United Arab Emirates University College of Sciences Department of Mathematical Sciences
FINAL EXAM
Complex Analysis I MATH 315 SECTION 01 CRN 23516 9:30 { 10:45 on Monday & Wednesday Date: Wednesday, January 6, 2010
ID No: 200
Name:
Score: /
1. (5 points) Express (1 + i
p 3)^9 in the form of x + iy.
2. (5 points) Use the real and imaginary parts of f(z) = ez^ to nd the real and imaginary parts
of g(z) = ef^ (z)^ = exp (ez^ ).
4. (5 points) Let S be the region [0; =3] [0; 1]. Find the image S^0 of S under the mapping
w = ezi^ and sketch the region S^0.
x
y
O
S
i =3 +^ i
5. (5 points) Let a function f be analytic everywhere in a domain D. Prove that if f is real{valued
for all z in D, then f must be constant throughout D.
6. (5 points) Prove that u(x; y) = log(x^2 + y^2 )^1 =^2 is harmonic on C f 0 g.
7. (5 points) Find the principal value of 1
(1 + i)^1 =^2
10. (5 points) Let C be the boundary of the triangle with vertices at the points 0, 3i and 4
oriented in the counterclockwise direction (see gure). Then show that Z C^ (e
z (^) z) dz 60 :
x
y
O
C
3 i
11. (5 points) Evaluate the contour integral
Z C z + 1 z = 2
! 3 dz, where C is the positively ori- ented circle jzj = 1.
12. (5 points) Evaluate the contour integral
Z C
ez^2 (z i)^3 dz, where^ C^ is the positively oriented circle jz ij = 1.
13. (5 points) Let f be an entire function such that jf(z)j 1 for all z. Prove that f is constant
on the whole complex plane.