The integral representation of Legendre functions is P_\nu(z) = \oint_{\Gamma} \frac{(w^2-1)^\nu}{(w-z)^{\nu+1}} dw. I'm trying to show that this satisfies Legendre's equation. When I take the derivatives and plug it into the equation, I just get a nasty expression with nasty integrals times...
First of all, thank you for the very informative post; it was very enlightening.
Why do sin(nx) form a complete basis in L^2([0,\pi]) ?
What is the essential difference? Does it have to do with odd and even functions?
Thanks again
Perhaps you could try using something that approaches a delta function like \frac{1}{\pi} \frac{sin(\lambda x)}{x} as \lambda \to \infty or maybe \sqrt{\frac{\alpha}{\pi}}e^{-\alpha x^2} as \alpha \to \infty
Perhaps you could find the residue w.r.t. \lambda and take the limit. Not...
So the other day in class my teacher gave a proof for the completeness of \phi_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx} in L^2([-\pi,\pi]) . And I'm trying to convince my self I understand it at least a little. He defined Frejer's Kernel
K_n(x) = \frac{1}{2\pi(n+1)}...
Hi, I'm looking for a good beginner text on Wavelets. Preferably an undergraduate or early graduate level. The background material that I'm missing most is infinite dimensional vector spaces/function spaces and Fourier analysis.
Any advice would really be appreciated!
Thanks!
Thanks for replying
Thats what I got for my residue, Sorry that z is just saying plug in e^i...
I try writing it in that form (a+bi), but its so messy!
What I get is
J=\frac{\pi\sqrt{2}}{2}e^{-a\frac{\sqrt{2}}{2}}(cos(a\frac{\sqrt{2}}{2})+sin(a\frac{\sqrt{2}}{2}))
Does anyone else think I'm...
Homework Statement
I'm trying to evaluate the integral
I(a)=\int\frac{cos(ax)}{x^{4}+1}
from 0 to ∞
Homework Equations
To do this I'm going to consider the complex integral:
J=\oint\frac{e^{iaz}}{z^{4}+1}
Over a semi-circle of radius R in the upper half plane, then let R-->∞...
I figured this out, should someone ever stumble upon this and be curious.
The singularities arise only due to the branch cuts! So any point on a branch but is a non-isolated singularity. However, the branch cut is somewhat arbitrary, so long as the cut ends at branch points. So the only points...
Homework Statement
Find all the singularities of
f(z)=log(1+z^{\frac{1}{2}})
Homework Equations
Well I need to expand this. Find if it has removable singularities, poles, essential singularities, or non-isolated singularities. The problem is the branches. I know z^{\frac{1}{2}}
has...
Well you should have
lim x-->∞ \frac{ln(1+2^{x})}{x}
Then use L'Hopitals rule.
You will find the limit of this. To get the answer you want you have to exponentiate it (since you took the natural log in order to find it).
Don't forget, that now
e^{y}=(1+2^{x})^{\frac{1}{x}}
So when you find y, the limit of
y=ln((1+2^{x})^{\frac{1}{x}})
you have to take e^{y} to get the answer to the limit you're looking for.