**Math 115 Methods of
Analysis and ApplicationsFall 2006**

Place: Science Center
411

Time: Tuesdays & Thursdays, 11:30 a.m. - 1 p.m.

**Instructor: Professor
Yum-Tong Siu ** (e-mail:
**siu@math.harvard.edu**)

**Office Hours: Tuesdays & Thursdays, 10 a.m. - 11:30
a.m., Science Center
511**

Textbooks:**
Lars Ahlfors: Complex Analysis, 3rd ed., McGraw-Hill 1978;
Walter A. Strauss: Partial Differential Equations. An Introduction.
John Wiley & Sons 1992 I. M. Gelfand & S. V. Fomin: Calculus of Variations, Dover 2000
**

Roughly two-fifths of the
Final Grade from the Final Exam, one-fifth from the First Mid-Term,

one-fifth
from the Second Mid-Term, and the remaining one-fifth from weekly homework
assignments

**Course Assistant:
Yan Zhang **** (e-mail:
yanzhang@fas.harvard.edu)
Sections: Fridays 4 - 5 p.m., Science Center 103b
Office Hours: Mondays 10 - 11 p.m., Quincy Dining Hall**

Syllabus: The course emphasizes methods and techniques and
their applications.

At the same time theory is introduced to provide
foundations for the methods and techniques.

Topics covered include:
Cauchy residue theory and its application to computation

of definite
integrals and the summation of series; conformal mappings; Fourier
analysis;

Hilbert spaces and operators; Laplace's equations; Bessel
and Legendre functions;

symmetries; Sturm-Liouville theory; calculus
of variations.

LECTURE NOTES: Divison of Complex Numbers, Square Roots, Higher-Order Roots, Multiple-Angle Formulae, and Trigonometric Sums

LECTURE NOTES: Complex Derivatives and Cauchy-Riemann Equations

LECTURE NOTES: Path Integration, Stokes's Theorem, and Cauchy's Theorem for Smooth Functions Satisfying the Cauchy-Riemann Equation

LECTURE NOTES: Integration of Rational Functions of Sine and Cosine

HOMEWORK: Homework Assignment Due September 28, 2006

CLICK HERE FOR YAN ZHANG'S SOLUTION OF HOMEWORK ASSIGNMENT #1

LECTURE NOTES: Theorem of Cauchy-Goursat and Cauchy's Integral Formula

LECTURE NOTES: Power and Laurent Series Expansion, Classification of Isolated Singularities, and Computation of Residues

HOMEWORK: Homework Assignment Due October 5, 2006

LECTURE NOTES: Definite Integrals Evaluated by Contour Integration Over a Half Circle

LECTURE NOTES: Definite Integrals Evaluated by Contour Integration of Branches of Holomorphic Functions

HOMEWORK: Homework Assignment Due October 17, 2006

DATE OF FIRST MID-TERM: NOVEMBER 2, 2006 (THURSDAY)

LECTURE NOTES: Conformal Mappings and Application to Electrostatics

HOMEWORK: Homework Assignment Due October 24, 2006

LECTURE NOTES: Applications of Conformal Mappings to Fluid Flow and Temperature Distribution

HOMEWORK: Homework Assignment Due October 31, 2006

LECTURE NOTES: Schwarz-Christoffel Transformations

CLICK HERE FOR YAN ZHANG'S SOLUTION OF HOMEWORK ASSIGNMENT #2

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LECTURE NOTES: Argument Principle

HOMEWORK: Homework Assignment Due November 7, 2006

CLICK HERE FOR YAN ZHANG'S SOLUTION OF HOMEWORK ASSIGNMENT #4

LECTURE NOTES: Derivation of the Poisson Kernel From the Use of the Argument Function

SOLUTIONS TO PROLBEMS IN FIRST MID-TERM EXAMINATION

HOMEWORK: Homework Assignment Due November 28, 2006

LECTURE NOTES: Derivation of the Poisson Kernel by Fourier Series and Convolution

LECTURE NOTES: Derivation of the Poisson Kernel From the Cauchy Formula

LECTURE NOTES: Maximum Principle

LECTURE NOTES: Derivation of the Poisson Kernel From the Method of Electrostatics

HOMEWORK: Homework Assignment Due November 28, 2006

LECTURE NOTES: Euler-Lagrange Equations for One Function of One Variable With Fixed End-Points and One Order of Differentiation

LECTURE NOTES: Brachistochrone

LECTURE NOTES: Catenary

LECTURE NOTES: Legendre Transformation

LECTURE NOTES: General Variation Formula and Weierstrass-Erdmann Corner Condition

LECTURE NOTES: Euler-Lagrange Equations for Many Functions and Variables and High-Order Derivatives

LECTURE NOTES: Canonical Transformation

HOMEWORK: Homework Assignment Due December 12, 2006

CLICK HERE FOR YAN ZHANG'S SOLUTION OF HOMEWORK ASSIGNMENT #5

CLICK
HERE FOR
YAN ZHANG'S SOLUTION OF HOMEWORK ASSIGNMENT #6

A Remark on Solution of Problem 2(b)

LECTURE NOTES: Legendre's Necessary Condition for Local Minimum

LECTURE NOTES: Conjugate Points and Sufficient Condition for Local Minimum

LECTURE NOTES: Lagrange Multipliers and Variational Problems with Constraints

LECTURE NOTES: Noether's Theorem on First Integrals From Transformations Leaving the Functional Invariant

HOMEWORK: Homework Assignment Due December 19, 2006

SOLUTIONS TO PROLBEMS IN SECOND MID-TERM EXAMINATION

LECTURE NOTES: Bessel Functions and Vibrating Circular Membrane

LECTURE NOTES: Legendre Functions and the Laplace Equation in Spherical Coordinates

HOMEWORK: Homework on Partial Differential Equations

DATE OF
FINAL EXAM: JANUARY 22, 2007 (MONDAY)

at 2:15 p.m. in Sever 107

CLICK
HERE FOR
YAN ZHANG'S SOLUTION OF HOMEWORK ASSIGNMENT #7

CLICK
HERE FOR
YAN ZHANG'S SOLUTION OF HOMEWORK ASSIGNMENT #8

CLICK
HERE FOR
YAN ZHANG'S SOLUTION OF HOMEWORK ASSIGNMENT #9

CLICK
HERE FOR
YAN ZHANG'S SOLUTION OF HOMEWORK ASSIGNMENT #10

SOLUTIONS TO PROLBEMS IN FINAL EXAMINATION