Dimensions of ##angle\cdot length^2## are correct. There are other factors of ##x_{\pm}^{-1},y_{\pm}^{-1},z_{\pm}^{-1},L^{-1}## and these lead to dimensions of ##[k]charge^2length^{-1}## when all is said and done. I just omitted those prefactors to focus on the integral I have not yet been able...
It is a schoolwork-type problem, but not schoolwork. I was investigating electrostatic properties of an analytically soluble charge distribution when I encountered these integrals.
Summary:: Could someone please evaluate this double integral over rectangular bounds? Answer only is just fine.
[Mentor Note -- thread moved from the technical math forums, so no Homework template is shown]
Hi,
I'm trying to find the answer to the following integral over the rectangle...
Hey all,
So this time I have a different kind of question - namely, "what is this called?"
I recall hearing/reading this in at least two places, one of which was YouTube. The idea is the following:
A RNG picks an integer uniformly from 1 to N. It picks 4. What is the expected value of N?
I'm...
No worries - It's a very tricky integral even (or perhaps especially?) without the Weierstrass substitution. :)
I have noticed that integrals arising from a physical scenario (rather than simply composed for mathematics) tend to have "nicer" answers. Without giving away too much, the integrals...
FYI I solved my problem on my own.
While my solution is perfectly valid, it uses a Weierstrass substitution which is actually not needed for the integral (although it is quite surprising that it is not necessary!). As luck would have it, the integral can be subdued without the sneaky...
Yes:
The formula you quote is just the inverse of the identity ##\tan(A+B)=(\tan A+\tan B)/(1-\tan A\tan B)##. While this seems an extremely obvious way to proceed, the expressions I get just become messier and do not begin to resemble the other side of the tentative equation I wish to prove...
Hello everyone,
I have a maths question (for a change). In summary, I would like to reconcile the following two integrals:
Integral A: https://www.wolframalpha.com/input/?i=integrate+(a^2tan^2theta)/(a-b+cos+theta)+dtheta
\int\frac{x^2\,dx}{\sqrt{x^2+a^2}(\sqrt{x^2+a^2}-b)}
=x...
mfb, I've edited the quote below to fill in the other pairs from my spreadsheet:
Also, if we no longer restrict to ##m,n<100##, then in fact there is an easy way to construct infinitely many of these pairs.
Since ##3^2+4^2=5^2## we have ##\frac{1}{20^2}+\frac{1}{15^2}=\frac{1}{12^2}##...
Wow, good on you guys for finding some working counterexamples!
Any chance you can provide the complete list of matches you found (including the trivial multiples)? I would like to see if the domain of validity of my partial proof was right - that is, any match must fall within one of the cases...
Hello all,
This is a problem of a different flavour from my usual shenanigans. I'm looking at a function
$$f(m,n)=\frac{m^2n^2}{(m+n)(m-n)}$$
and am trying to determine if there are any two pairs of values ##(m_1,n_1)## and ##(m_2,n_2)## which evaluate to the same result. Assume that...
A reflectionless potential?!
See, it's posts like yours that make me love PhysicsForums. I've studied a lot of interesting things that branch off of basic physics ideas - the Capstan equation, Maxwell's fish-eye lens, and this problem of late - and it's little gems like what you've just shared...
Hi Jason and Hutch,
Thank you both for your in-depth responses.
My main problem with this approach was simply that I was expecting the derivation to break when we did not assume a uniform medium. This was true in my freshman mechanics derivation for transverse waves on a string. However, I...