6/6/2010
Special functions
Special functions independent study
Reading, assignments, and due dates for assignments are listed below.
Throughout, AW refers to Arfken and Weber, sixth edition.
Assignment
Due Date
1
Fri May 21
2
Fri May 28
3
Fri June 4
4
Fri
In class we showed that if f (z ) is a holomorphic function with simple isolated zeroes located
at a1 , a2 , , none of the ai zero, then f (z ) could be written in terms of f (0), f (0), and the
ai as
z
f (0)
1
f (z ) = f (0) exp z
exp(z/an )
f (0) n
an
1
1. An example of convolution
Consider a function
f (x) = cos x + cos 3x
1
1
(exp(ix) + exp(ix) + (exp(3ix + exp(3ix)
=
2
2
Imagine passing this function through a low-pass lter, that multiplies the coecients of 1,
exp(ix) by 1, and multiplies all other Fo
Gegenbauer polynomials
1
Denition
Generating function:
1
=
(1 2xt + t2 )
(
Cn) (x)tn
n=0
for = 0.
(
The Gegenbauer polynomials Cn) (x) are also known as ultraspherical polynomials (see
Arfken-Weber end of section 12.1).
The Gegenbauer polynomials include
Elliptic functions
See [1][section 4.5] and [2] for more information.
1
Denition
An elliptic function is a single-valued doubly-periodic function of a single complex variable
which is analytic except at poles and whose only singularities in the nite plane
Appendix D
Asymptotic series
Asymptotic series play a crucial role in understanding quantum eld theory, as Feynman diagram
expansions are typically asymptotic series expansions. As I will occasionally refer to asymptotic
series, I have included in this ap
Appendix C
Integral transforms
C.1
Fourier transform
Given a real-valued function f (x) on the real line, dene the Fourier transform of f (x) to be
f (k) =
f (x) exp(ikx)dx
(C.1)
dk
f (k) exp(ikx)
2
(C.2)
Then it can be shown that
f (x) =
known as the in
Riemann Surfaces
See Arfken & Weber section 6.7 (mapping) or [1][sections 106-108] for more information.
1
Denition
A Riemann surface is a generalization of the complex plane to a surface of more than one
sheet such that a multiple-valued function on the
Analytic Continuation
See Arfken & Weber pp 432-434 (in section 6.5 on Laurent expansions) for some of the
material below. Our description here will closely follow [1].
1
Denition
The intersection of two domains (regions in the complex plane) D1 , D2 , de
Syllabus, Fall 2009
HISTORY OF ECONOMIC THOUGHT
W. A. Schaffer
Econ 4620A
Fall 2009
MWF 2, IC 109
TEXTS: (1) Todd G. Buchholz, New Ideas from Dead Economists, Revised
Edition, New York: Plume (Penguin Putnam, Inc.), 1999. (2) Robert L.
Heilbroner, The Wor
From Russian Site:
1. R. Ekelund, R.Herbert A History of Economic Theory and Method. McGraw Hill Publ. Co. 1990.
2. M. Blaug Economic Theory in Retrospect Cambridge University Press 1991
3. T. Negishi History of Economic Theory North-Holland 1989
4.
ECONOMICS 4177: HISTORY OF ECONOMIC THOUGHT
Fall 2009
COURSE:
MW 12:30-1:45
Davis
History of Economic Thought
COURSE DESCTIPTION: ECON 4177. History of Economic Thought. (W) (3) (3G)
Prerequisites: ECON 1201 and 1202. History of economics as a science and