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Imaginary Parts - Complex Analysis - Exam, Exams of Mathematics

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

Uploaded on 02/12/2013

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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:
/70
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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.