MATH 113 Complex Analysis
Spring 2024

Place: Science Center 309A
Time: Tuesdays, Thursdays, 12 noon - 1:15 p.m.

Instructor: Professor Yum-Tong Siu   (e-mail: siu@math.harvard.edu)
Office Hours: Tuesdays, Thursdays 10:30 a.m. - 12 noon, Science Center 511

Textbook: Elias M. Stein & Rami Shakarchi: Complex Analysis, Princeton University 2003

Course Assistants:
Daniel Sheremeta (e-mail: danielsheremeta@college.harvard.edu)
Office Hours: Mondays, 8 - 9 p.m., Leverett Dining Hall
Hahn Lheem (e-mail: hahnlheem@college.harvard.edu)
Eric Yan (e-mail: eyan@college.harvard.edu)
Office Hours: Hahn Lheem and Eric Yan jointly Wednesdays, 8 - 10 p.m., Mather Dining Hall

Syllabus:  Analytic functions of one complex variable: power series expansions, contour integrals, Cauchy's theorem, Laurent series and the residue theorem. Some applications to real analysis, including the evaluation of indefinite integrals. An introduction to some special functions.

Final Grade:  Computed from weekly homework assignments, one take-home midterm (worth 3 homework assignments), and the take-home final (worth 6 homework assignments), with allowance for marked improvements toward the end of the course.

LECTURE NOTES: (Lecture 1) Complex Numbers

LECTURE NOTES: (Lecture 2) Complex Differentiation and Cauchy-Riemann Equations

HOMEWORK: (Assignment #1) Assigned January 25, 2024 due February 1, 2024

LECTURE NOTES: (Lecture 3) Holomorphic Functions Defined by Power Series and Exponential and Sine and Cosine Functions as Entire Functions

LECTURE NOTES: (Lecture 4) Theorem of Cauchy-Goursat

HOMEWORK: (Assignment #2) Assigned February 1, 2024 due February 8, 2024

LECTURE NOTES: (Lecture 5) Cauchy's Integral Formula and Power and Laurent Series Expansion

LECTURE NOTES: (Lecture 6) Characterization of Poles and Essential Singularities and the Residue Theorem

LECTURE NOTES: (Lecture 7) Evaluation of Denite Integrals by Residues without Using Branches of Holomorphic Functions

HOMEWORK: (Assignment #3) Assigned February 8, 2024 due February 15, 2024

LECTURE NOTES: (Lecture 8) Evaluation of Denite Integrals by Residues and the Use of Branches of Holomorphic Functions

HOMEWORK: (Assignment #4) Assigned February 15, 2024 due February 22, 2024

LECTURE NOTES: (Lecture 9) Some Useful Theorems in Complex Analysis as Consequences of Cauchy's Integral Formula and Power Series Expansion

LECTURE NOTES: (Lecture 10) Computation of Infinite Sums by Residues and Partial Fraction Expansion of Meromorphic Functions

HOMEWORK: (Assignment #5) Assigned February 22, 2024 due February 29, 2024

LECTURE NOTES: (Lecture 11) Argument Principle and Rouché's Theorem from Residue Computation of Logarithmic Derivative

HOMEWORK: (Assignment #6) Assigned February 29, 2024 due March 7, 2024

LECTURE NOTES: (Lecture 12) Holomorphicity Characterized by Conformality and Preservation of Orientation

Closed-Book Take-Home Midterm Test (Click to Download)

To be taken in any 4-hour period of your choice between 12:01 a.m. March 18, 2024 and 11:59 p.m. March 20, 2024.

Submit your solution PDF file to CANVAS website for Math 113.

Do not read the problems unless you are ready to start working on them in your chosen 4-hour period.

SOLUTION (with Grading Rules) of Take-Home Midterm Test (Click to Download)

LECTURE NOTES: (Lecture 13) Linear Fractional Transformations and Biholomorphisms of Extended Gauss Plane and Open Unit Disk

LECTURE NOTES: (Lecture 14) Application to Fluid Flow, Temperature Distribution, Electrostatic Potential and Airfoil Lift

HOMEWORK: (Assignment #7) Assigned March 21, 2024 due March 28, 2024

For Assignment #7 and all homework assignments after #7, there will be a deduction
of 10% for each day of late submission in the computation of the final grade

LECTURE NOTES: (Lecture 15) The Riemann Mapping Theorem

LECTURE NOTES: (Lecture 16) Boundary Behavior of Biholomorphism of Piecewise Smooth Domain

LECTURE NOTES: (Lecture 17) Schwarz-Christoffel Transformations

HOMEWORK: (Assignment #8) Assigned March 28, 2024 due April 4, 2024

LECTURE NOTES: (Lecture 18) Poisson Kernel

HOMEWORK: (Assignment #9) Assigned April 4, 2024 due April 11, 2024

LECTURE NOTES: (Lecture 19) Elliptic Functions

HOMEWORK: (Assignment #10) Assigned April 11, 2024 due April 18, 2024

HOMEWORK: (Assignment #11) Assigned April 18, 2024 due April 25, 2024

LECTURE NOTES: (Lecture 20) Theorem of Gelfond-Schneider on Transcendental Numbers

Closed-Book Take-Home Final Exam (Click to Download)

To be taken in any two 4-hour periods (separated by a 2-hour intermission) of your choice
between 12:01 a.m. April 29, 2024 and 11:59 p.m. May 5, 2024.
Time for typing up your file in TeX is extra and does not count toward the total 8 hours of your choice.

Submit your solution PDF file to CANVAS website for Math 113.

Do not read the problems unless you are ready to start working on them in your chosen two 4-hour periods.