Homework Statement
I put this in the math forum because although it's for my EM waves class, it's a math question.
Show that the spin force can be written as:
F_{spin}=\frac{-1}{2}Im(\alpha)Im(E\cdot\nabla E^{*})=\nabla\times L_s
Find L_s.
Where \alpha is complex. I'm using E^{*} to denote...
LCKurtz,
I can't find anything like that in my textbook. Did you do that so the solution would satisfy both the nonhomogenous and homogenous parts? I suppose you added the extra function of x because the nonhomogenous boundary conditions only depend on x?
Homework Statement
Solve the diffusion equation:
u_{xx}-\alpha^2 u_{t}=0
With the boundary and initial conditions:
u(0,t)=u_{0}
u(L,t)=u_{L}
u(x,0=\phi(x)
The Attempt at a Solution
I want to solve using separation of variables...
I start by assuming a solution of the form...
I figured out what I was doing wrong. I realize now the substitutions I was making from the Euler formula don't work over the real line, those are more suited for integration over the unit circle. Once I chose a better way to transform into an integral in Z, this turned out to be a very simple...
Okay, taking your advice.
For the example problem,
\int^{∞}_{-∞} \frac{e^{it}}{1+t^2}
We can directly substitute t=z
\oint ^{∞}_{-∞} \frac{e^{iz}}{1+z^2} = \oint ^{\infty}_{-\infty} \frac{e^{iz}}{(1+i)(1-i)}
Now since it's clear to see that f(z)→0 as R→∞, we can integrate over the...
Right, this acts like a 'sinc' function, so the Jordan contour is the right selection for integration.
So the function is:
f(t)=\frac{cos(\alpha t)}{\beta^2+t^2}
Let:
z=e^{i\alpha t}
dz=i\alpha e^{i\alpha t}dt
cos(\alpha t)=\frac{z+z^{-1}}{2}
Transforming f(t) to f(z)...
Yes I've used the residue theorem to evaluate contour integrals. We've covered so much in this class I had already forgotten that that technique would work.
Since this involves a 2\pi periodic function, I'd integrate over the unit circle in the complex plane... Use Euler's formula for the...
Homework Statement
Evaluate the integral:
\int^{2\pi}_{0}\frac{d\theta}{(A+Bcos(\theta))^2}
a^2>b^2
a>0
The Attempt at a Solution
First, I convert this to contour integration along a full sphere in the complex plane.
I let:
z=e^(i\theta)
dz=ie^(i\theta)
d\theta=-idz/z...
Homework Statement
Can a function which is purely real-valued be analytic? Describe the behavior of such functions?
Homework Equations
The Cauchy-Riemann conditions
ux=vy, vx=-uy
The Attempt at a Solution
I can't think of any pure real-valued equations off the top of my head which...