Recent content by jjr

I ended up going a different route. I was working with the ##\text{tanh}(t)## function because I needed something that closely resembles the error function ##\text{erf}(t)=\frac{1}{\sqrt{\pi}}\int_{-t}^t e^{-t'^2}dt'##, for which I initially couldn't find the F.T. I managed to calculate it after...

Sorry, I should of course have added that running it through WolframAlpha returns ##\mathcal{F}_t[\text{tanh}(t)](\omega) = i\sqrt{\frac{\pi}{2}}\text{csch}\left(\frac{\pi\omega}{2}\right)## which I'm hoping to obtain analytically

Hello I am trying to determine the Fourier transform of the hyperbolic tangent function. I don't have a lot of experience with Fourier transforms and after searching for a bit I've come up empty handed on this specific issue. So what I want to calculate is: ##\int\limits_{-\infty}^\infty...

That was very helpful! Thank you

Hi! A friend of mine is interested in a PhD position in a Scandinavian country. There are no specific calls for projects, but the group webpage mentions available positions and encourages anyone that is interested to send an email for information. A formal application obviously comes later, so...

Homework Statement Describe all the singularities of the function ##g(z)=\frac{z}{1-\cos{z}}## inside ##C## and calculate the integral ## \int_C \frac{z}{1-\cos{z}}dz, ## where ##C=\{z:|z|=1\}## and positively oriented. Homework Equations [/B] Residue theorem: Let C be a simple closed...

Thanks! So if I understand correctly: ##|z^4-z^3+z^2-z| \leq |z^4| + |z^3| + |z^2| + |z| \leq 4 ## because ##|z^n|<1## if ##|z|<1##, where ## n \in {1,2,3,4} ##. And because ##8<9## it's proven.J

Homework Statement Let ##D={z : |z| <1}##. How many zeros (counted according to multiplicty) does the function ##f(z)=2z^4-2z^3+2z^2-2z+9## have in ##D##? Prove that you answer is correct. Homework Equations 3. The Attempt at a Solution [/B] The function has no zeros in ##D##, which can be...

The first

Thank you, got it now! ## \lim_{i\to -1} \frac{\sqrt{z}-1+\sqrt{z+1}}{\sqrt{z^2-1}} ## ##\to \lim_{i\to -1} \frac{\sqrt{z}-1+\sqrt{z+1}}{\sqrt{(z+1)(z-1)}} ## ##\to \lim_{i\to -1} \frac{1}{\sqrt{z-1}} = \frac{1}{i\sqrt{2}} ##

Homework Statement Calculate the following limit if it exists ## lim_{z\to -1}\frac{\sqrt{z}-i+\sqrt{z+1}}{\sqrt{z^2-1}} ## the branch of root is chosen to that ##\sqrt{-1}=i## Homework Equations 3. The Attempt at a Solution [/B] By inserting ##z=-1## directly, I get a ##\frac{0}{0}##...

Yeah, it seems you are correct. Thanks!

Hi I'm converting some numbers to atomic units, and it's important that I get it right. Would be happy if someone could reassure me. I want to convert the units of an electric field measured in kV/cm to atomic units. Here is what I got: \frac{kV}{cm}=\frac{10^3}{10^{-2}}\frac{J}{C\cdot...

Sorry, made a mistake. Edited now.

## (z-i)(z^2+iz-1) = z^3 + iz^2 - z -iz^2 -i^2z+i = z^3+i ##