SEMINARIO
I PRO E CONTRO DELLITALICUM *
A cura di Alessandro Gigliotti
SOMMARIO: 1. Alessandro Gigliotti, Introduzione. 2. Fulco Lanchester, Innovations institutionnelles et sparation des pouvoirs:considrations sur le
dangereux chevauchement des diffrent

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
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
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
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
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
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
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

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Obiettivo specifico 3.1 Rilancio della propensione agli investimenti del sistema produttivo
Azione 3.1.1 Aiuti per investiment

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WALKING COLUMN DETAIL (ALTERNATE)
S-2
SCALE: 1/4" = 1'-0"
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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
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
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
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
Homework Assigned on November 9, 2006
due November 28, 2006
Problem 1 (Poissons Integral Formula for the Upper Half-Plane [#1 on p.171
of Ahlfors]). Verify Part (a) by imitating the derivation of the Poisson integral fo

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