Lahore University of Management Sciences
Math 300 – Complex Variables
Spring 2016-2017
Instructor
Room No.
Office Hours
Email
Telephone
Secretary/TA
TA Office Hours
Course URL (if any)
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Shazia Zafar & Noreen Sohail /
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Course Basics
Credit Hours
Lecture(s)
Recitation/Lab (per week)
Tutorial (per week)
4
Nbr of Lec(s) Per Week
Nbr of Lec(s) Per Week
Nbr of Lec(s) Per Week
2 Lectures per week
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Duration
Duration
Duration
100 minutes each
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COURSE DESCRIPTION
The theory of functions of a complex variable is a rewarding branch of mathematics to study at the undergraduate level
with a good balance between general theory and examples. Topics include the complex number system, analytic
functions and their properties, further analytic functions including exponential, logarithm, trigonometric and hyperbolic
functions of a complex variable, Cauchy‐Riemann equations, complex integration and contour integrals, Cauchy‐Goursat
theorem, Cauchy’s integral formula, Taylor series and Laurent series, the calculus of residues and poles, improper
integrals, and conformal mappings.
COURSE PREREQUISITE(S)
•
Math 3010 (Advanced Calculus) OR Math 301 (Real Analysis I) OR Math 205 (Introduction to Analysis I)
COURSE OBJECTIVES
At the end of the course student should have familiarity with the basics of complex analysis and learn a number
of powerful techniques (for example, evaluation of integrals) with applications in many aspects of both
pure and applied mathematics, and other disciplines, particularly the physical sciences.
Learning Outcomes
Lahore University of Management Sciences
1
2
3
Complex differentiation and integration
Series and convergence
Conformal mappings
Grading Breakup and Policy
Quiz(s)/Assignment:
20%
Class Participation:
Attendance:
Midterm Examination: 35%
Project:
Final Examination:
45%
Examination Detail
Midterm
Exam
Yes/No:
Combine Separate:
Duration:
Preferred Date:
Exam Specifications:
Yes
Separate
2 hours
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Final Exam
Yes/No:
Combine Separate:
Duration:
Exam Specifications:
Yes
Separate
2 hours
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COURSE OVERVIEW
Lecture/
Topics
Module
1)
Properties of complex numbers
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
Exponential form and roots
Functions and Mappings
Limits
Continuity
Differentiable and analytic functions
The Cauchy ‐ Riemann equations
Harmonic functions
Exponential and logarithmic functions
Branches and derivatives of Logarithms
Trigonometric and Hyperbolic functions
Inverse Trigonometric and Hyperbolic
functions integrals
Complex
Contours and Contour integrals
Cauchy‐Goursat theorem
Cauchy’s integral formula
Recommended
Readings
Objectives/
Application
Lahore University of Management Sciences
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
Liouville’s theorem
Taylor series
Laurent series
Residues and Cauchy’s residue theorem
Zeros and poles
Evaluation of improper integrals
Indented paths
Integration along a branch cut
Improper integrals involving trigonometric
functions
Linear
mappings
Mappings of upper half plane
1
Conformal mappings
Textbook(s)/Supplementary Readings
Text: Complex Variables and applications, 9th edition, by J.W. Brown and R. V. Churchill, Published by
McGraw-Hill Education, 2013.
References:
Functions of One Complex Variable I, 2nd edition, by John B. Conway, Published by Springer, 1978
Visual Complex Analysis by Tristan Needham, Published by Clarendon Press; Reprint edition (February 18, 1999)