Math 115 Methods of
Analysis and Applications
Fall 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
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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
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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
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HERE FOR
YAN ZHANG'S SOLUTION OF HOMEWORK ASSIGNMENT #7
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HERE FOR
YAN ZHANG'S SOLUTION OF HOMEWORK ASSIGNMENT #8
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HERE FOR
YAN ZHANG'S SOLUTION OF HOMEWORK ASSIGNMENT #9
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HERE FOR
YAN ZHANG'S SOLUTION OF HOMEWORK ASSIGNMENT #10
SOLUTIONS TO PROLBEMS IN FINAL EXAMINATION