3
votes
1answer
137 views

Why study Bergman Spaces?

I'm interested in Operator Algebras and mathematical physics; recently, a friend showed me Duren and Schuster's "Bergman Spaces". I've read about two chapters now and I see there is a nice play ...
1
vote
0answers
30 views

Quantization and evaluating a complex function on multiple Riemann sheets

In Lecture 3 of his mathematical physics course, Carl Bender mentions that the evaluation of a complex function on multiple Riemann sheets can be used to describe the quantization of the laws of ...
0
votes
1answer
82 views

Conformal Mapping preserves the angles between the smooth curves.

I am interested to proof that conformal mapping preserves the angles between the smooth curves. I will be greatful if anyone can help me in it.
1
vote
1answer
56 views

Scalar Product Conditions

Let $x$ and $y$ be two vectors, $x\cdot y$ their scalar product, $\beta$ the angle between the vectors, and $|x|$ and $|y|$ their absolute values. Then we have $$|x| |y| \cos \beta =x \cdot y \quad ...
0
votes
0answers
63 views

Representing real function as integral over trigonometric functions

Since one can clearly express any function g(x) as integral from 0 to infinity of A(k)cos(kx)dk + integral from 0 to infinity of B(k)sin(kx)dk, how would G(k) relate to A(k) and B(k)? In other words, ...
0
votes
0answers
100 views

Passing the singularities .

I need some information or detail with example to the following statements. Circumvent the singularity by a contour inside the wave-guide. Circumvent the singularity by a contour outside the ...
2
votes
1answer
40 views

Laurent series of the function $M(s) = \dfrac{(1-s)[\Gamma(s)\Gamma(1-s)]^2}{(2+s)(1+2s)(3+2s)} $

The function has double poles at $ s = 2,3,\ ... $ (and at other points as well but I am interested only at these points.) Its given that the principal part of Laurent series at these points $ s = n, ...
2
votes
2answers
258 views

Contour Integrals and Residues

I'm trying to figure out what it is all about, but my mind is blowing up. First of all, I have turned back and looked at the general definitions of integrals. Then I have looked to line integrals. ...
1
vote
1answer
158 views

Finding the integral $\int_0^\pi\dfrac{d\theta}{(2+\cos\theta)^2}$ by complex analysis

Trying to find the integral $\int_0^\pi\dfrac{d\theta}{(2+\cos\theta)^2}$ by complex analysis, I let $z = \exp(i\theta)$, $dz = i \exp(i\theta)d\theta$, so $ d\theta=\dfrac{dz}{iz}$. I am trying ...
1
vote
2answers
576 views

finding Laurent series for $\dfrac{1}{z(z-2)^3}$

I am trying to get the Laurent series for $\dfrac{1}{z(z-2)^3}$. I know there are poles at $z = 0$ and $z=2$, and so I am looking for expansions about the singularities. Using $\dfrac{1}{1-z} = ...
0
votes
1answer
359 views

Laurent Series and Taylor Series

I am trying to find the Laurent series of $\dfrac{1}{(1+x)^3}$; would this be the same as finding the Maclaurin series for the same function?
2
votes
1answer
66 views

Raise a Laurent series to a power

I want to know if there is a general rule by which a Laurent series can be raised to a power; in other words,I have the Laurent series for $f(z) =\dfrac{1}{z(z-1)(z-2)}$ about a certain pole, and I ...
2
votes
2answers
270 views

integrating $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)}dz$ on $|z|=4$

I am doing $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)}dz$ on $|z|=4$ and I find that there are poles within the contour at $z = 1$ and at $z = 3i$, both simple poles. I find that the integral $I = 2\pi ...
1
vote
3answers
218 views

finding residue with $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)} dz$

I am doing the integral $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)} dz$, and I am trying to find the residue at the pole $3i$;I am unsure how to do this. Could I factor $z^2 + 9$ further?
1
vote
1answer
124 views

How to expand a fraction in powers of $z$ or $\dfrac{1}{z}$, and which to do, in determining Laurent series

I have a function $f(z)=\dfrac{12}{z(2-z)(1+z)}$, I'm trying to find the Laurent series for each of the three annuli. The singularities are at $z = 0$, $z = 2$, and $z = -1$, so I'm looking for three ...
2
votes
0answers
127 views

Show Smoothness by Morera

I'm trying to show smoothness on $(0,\infty)(\Re)$ of the following function: $$ f(t,x)=\sum_{n=-\infty}^\infty e^{-\large \frac{(x-2\pi n)^2}{2t}}\frac{1}{\sqrt{2\pi t}} $$ The function is ...
1
vote
1answer
120 views

Integration of a potential over a cylinder.

It already happened several times. I'm trying to calculate a field, created by something-distributed-over infinite cylinder. And I get an integral like this: ...
0
votes
2answers
578 views

integrating with infinite-radius semicircle

Sorry, I'm really confused on this one.... My math physics class is having us evaluate closed contours, where the contour is on the real line and at the infinities we turn towards the upper half of ...