Math 115 (2006-2007) Yum-Tong Siu
1
Divison of Complex Numbers, Square Roots, Higher-Order Roots,
Multiple-Angle Formulae, and Trigonometric Sums
(1) To divide +i by +i, we make the denominator root by multiplying
both the numerator and the denominator by

Math 115 (2006-2007) Yum-Tong Siu
1
Definite Integrals Evaluated by Contour Integration Over a Half
Circle.
We are going to use Cauchys residue theory over the boundary of a half
disk to evaluate definite integrals of the following form.
Z
P (x)
dx with

Math 115 (2006-2007) Yum-Tong Siu
1
Derivation of the Poisson Kernel
From the Use of the Argument Function
.
We are going to derive the Poisson kernel by using the argument function
and plane Euclidean geometry. We start out with the basic building block

Math 115 (2006-2007) Yum-Tong Siu
1
Euler-Lagrange Equations for One Function
of One Variable With Fixed End-Points
and One Order of Differentiation
Context. Fix < a < b < and A, B R. Fix a continuously differentiable function F = F (x, y, y ) of three in

Math 115 (2006-2007) Yum-Tong Siu
1
Derivation of the Poisson Kernel
From the Cauchy Formula
.
The Cauchys integral formula states that
Z
1
f ()d
f (z) =
2i |=1 z
for |z| < 1 if f is holomorphic on the open unit 1-disk and continuous up to
the boundary. W

Math 115 (2006-2007) Yum-Tong Siu
1
Euler-Lagrange Equations for Many Functions
and Variables and High-Order Derivatives
The Case of High-Order Derivatives. The context is to find the extremal for
the functional
Z b
F x, y, y , , y (k) dx
x=a
at the funct

Math 115 (2006-2007) Yum-Tong Siu
1
Homework Assigned on October 19, 2006
due October 31, 2006
Problem 1. Let T denote the bounded steady temperature on the first quadrant cfw_x > 0, y > 0 with the following three constraints:
(i) The boundary value of T

Math 115 (2006-2007) Yum-Tong Siu
1
Maximum Principle
Let u be a real-valued harmonic function on a domain in C. One
property which a harmonic function enjoys is the mean-value property which
states that
Z 2
1
u a + rei d
()
u (a) =
2 =0
whenever cfw_ |z

Math 115 (2006-2007) Yum-Tong Siu
1
Homework Assigned on December 7, 2006
due December 19, 2006
Problem 1 (#18 on Page 52 of Gelfand and Fomin). Find the extremals of
the functional
Z 1
2
2
y + x dx,
J[y] =
0
subject to the conditions
y(0) = 0,
y(1) = 0,

Math 115 (2006-2007) Yum-Tong Siu
1
Brachistochrone (Curve of Shortest Time)
Statement of the Problem. The problem is to find a curve from (x, y) =
(0, h) to some point on the vertical line x = a such that when a particle
sliding down the curve by gravity

Math 115 (2006-2007) Yum-Tong Siu
1
Argument Principle
The argument principle helps us determine the number of zeroes of a
holomorphic function on the domain enclosed by a curve in its domain of
definition. To introduce it, let us start out with the well-

Math 115 (2006-2007) Yum-Tong Siu
1
Derivation of the Poisson Kernel by
Fourier Series and Convolution
.
We are going to give a second derivation of the Poisson kernel by using
Fourier series and convolution. We also use this derivation as an opportunity

Math 115 (2006-2007) Yum-Tong Siu
1
Homework Assigned on October 26, 2006
due November 7, 2006
Problem 1 (fluid flow in a channel through a slit). Let Q be a positive
number. (a) Find a bounded harmonic function u on the upper half-plane so
that
(i) its b

Math 115 (2006-2007) Yum-Tong Siu
1
Legendre Transformation
Two Ways to Define a Plane Curve. Usually a plane curve in the xy-plane
is defined by y = f (x). This way is to define it as the locus of a point
(x, y) subject to the constraint y = f (x). There

Math 115 (2006-2007) Yum-Tong Siu
1
Homework Assigned on November 16, 2006
due November 28, 2006
Problem 1 (Poissons Integral Formula for the Exterior of a Circle [#4 on
p.167 of Strauss]). Suppose u is a bounded twice continuously differentiable
function

Math 115 (2006-2007) Yum-Tong Siu
1
Homework Assigned on November 30, 2006
due December 12, 2006
Problem 1 (#5 on p.64 of Gelfand and Fomin). Find the curves for which
the functional
Z x1 p
1 + y2
J[y] =
dx, y(0) = 0
y
0
can have an extrema if
(a) the poi

Math 115 (2006-2007) Yum-Tong Siu
1
Lagrange Multipliers and Variational Problems with Constraints
Integral Constraints. Consider the variational problem of finding the extremals for the functional
Z b
F (x, y, y ) dx
J[y] =
a
with y(a) = A and y(b) = B s

Math 115 (2006-2007) Yum-Tong Siu
1
Legendre Functions and the Laplace Equation
in Spherical Coordinates
Dirichlet Problem on the Unit 3-Dimensional Unit Ball. We are going to
introduce the (associated) Legendre functions by considering the solution of
th

Math 115 (2006-2007) Yum-Tong Siu
1
Theorem of Cauchy-Goursat and Cauchys Integral Formula
Differentiable Functions Satisfying Cauchy-Riemann Equation Equivalent to
Complex Differentiable. Recall that a real-valued function of two real variables g(x, y) i

Math 115 (2006-2007) Yum-Tong Siu
1
Math 115 Final Examination with Solutions
January 22, 2007, 2:15 p.m. - 5:15 p.m.
Sever 107
Some formulae are provided at the end of the problem list. Not all of them
are necessary for solving the problems.
Problem 1. U

Math 115 (2006-2007) Yum-Tong Siu
1
Applications of Conformal Mappings to
Fluid Flow and Temperature Distribution
Fluid Flow. Denote by ~v = (p, q) the velocity of a steady 2-dimensional fluid
flow in a domain .H (Steady
means time-independent.) The flow

Math 115 (2006-2007) Yum-Tong Siu
1
Conformal Mappings and Application to Electrostatics
We will apply holomorphic functions (conformal mappings) to the problem of finding electrostatic potentials with prescribed constant boundary values for the two dimen

Math 115 (2006-2007) Yum-Tong Siu
1
Homework Assigned on September 21, 2006
due September 28, 2006
Problem 1 (from Ahlfors p.2, #1 and p.4, #1). Compute the real part and
the imaginary part of the following two complex numbers in terms of rational
numbers

Math 115 (2006-2007) Yum-Tong Siu
1
Noethers Theorem on First Integrals From
Transformations Leaving the Functional Invariant
Consider the variation problem of the functional
Z x2
F (x, y, y ) dx.
J=
x1
Assume that we have a 1-parameter family of transfor

Math 115 (2006-2007) Yum-Tong Siu
1
Conjugate Points and Sufficient Condition for Local Minimum
We now discuss a sufficient condition for an extremal function to be a local
minimum by using conjugate points. We consider the variation problem in
the simple

Math 115 (2006-2007) Yum-Tong Siu
1
Bessel Functions and Vibrating Circular Membrane
Method of Separation of Variables. For a linear partial differential equation
Lu = 0, we can use the method of separation of variables when the linear
partial differentia

Math 115 (2006-2007) Yum-Tong Siu
1
Homework Assigned on October 5, 2006
due October 17, 2006
Problem 1 (partly from Ahlfors p.161, #3). Evaluate the following integrals
by the method of residues:
Z
x2 x + 2
(a)
dx
4
2
x + 10x + 9
(b)
Z
0
(c)
Z
x sin x

Math 115 (2006-2007) Yum-Tong Siu
1
Complex Derivatives and Cauchy-Riemann Equations
Complex Derivative. Let z = x + iy be the coordinate of the complex plane
C and let f (z) be a complex-valued function on some open neighborhood U
of some point c = a + i

Math 115 (2006-2007) Yum-Tong Siu
1
Partial Fraction Expansion of Meromorphic Functions, Infinite Product Expansion of Entire Functions, and Summation of Series by
Residues and the Cotangent Function
Partial Fraction Expansion of Meromorphic Functions. Su

Math 115 (2006-2007) Yum-Tong Siu
1
Catenary (Hanging Cable)
Statement of the Problem. The problem is to find the equation of a hanging
cable of uniform density whose two end-points are fixed.
Writing Down the Functional From Minimizing the Potential Ener

Math 115 (2006-2007) Yum-Tong Siu
1
Definite Integrals Evaluated by Contour Integration of Branches of
Holomorphic Functions.
We are going to discuss the evaluation of definite integrals which requires
the application of Cauchys residue theory to branches

Math 115 (2006-2007) Yum-Tong Siu
1
Homework Assigned on September 28, 2006
due October 5, 2006
Problem 1 (from Ahlfors p.108, #2, #7; p.120, #1, #3). (a) Let r > 0 and
x be the real part of the complex variable z. Compute
I
x dz
|z|=r
for the positive se

Math 115 (2006-2007) Yum-Tong Siu
1
Schwarz-Christoffel Transformations
Schwarz-Christoffel transformations are used to map the upper half-plane
H := cfw_y > 0 to a given polygon which may be bounded or unbounded. An
unbounded polygon simply means a domai

Math 115 (2006-2007) Yum-Tong Siu
1
Power and Laurent Series Expansion, Classification of Isolated Singularities, and Computation of Residues.
We are going to take the next step into complex analysis from the practical
viewpoint of the computation of defi

Math 115 (2006-2007) Yum-Tong Siu
1
Homework Assigned on October 12, 2006
due October 24, 2006
Problem 1. (a) Verify the following infinite partial fraction decomposition
X
1 1
1
1
= + 2z
.
z
2
22
e 1
z 2
z
+
4n
n=1
(b) Verify the following infinite parti