The Proof is Trivial!

How can you evaluate that integral though?

How can you evaluate that integral though?

Doesn't seem impossible, right? It's e^z around the unit semicircle. Currently practically asleep, though, so I might be talking nonsense.

Surely then the integral would be 0 (no poles), which is not correct?

(I haven't done contour integrals in over a year btw.)

I can't be bothered to dig out some of my notes, so this may not be rigorous:

It's e^z around the unit semicircle, which isn't a closed loop. We just need to evaluate the integral along the x-axis to complete the semicircular closed contour: $-\int_{-1}^1 e^z dz$. OK, I've definitely got something horribly wrong :P

Solution 483

This is an elementary solution to the problem (here meaning not using complex analysis). For full disclosure, I used ThatPerson's hint.

I'm not sure that . I got a different answer for that part, though it doesn't affect the end since we're only interested in the real part.

Also, apparently this was set in a Cambridge exam over a hundred years ago:

Problem 484**

$\displaystyle \int^{4}_{0} \dfrac{\ln (x)}{\sqrt{4x-x^2}} \ dx$

Is it ?

I'll type up my "solution" tomorrow if I'm right, I've only done a little probability though so the chances are I'm far off.

I'm not sure that . I got a different answer for that part, though it doesn't affect the end since we're only interested in the real part.

Also, apparently this was set in a Cambridge exam over a hundred years ago:

Problem 484**

$\displaystyle \int^{4}_{0} \dfrac{\ln (x)}{\sqrt{4x-x^2}} \ dx$

I get my answer as:

Pretty average tbh

Problem 471
Determine all sets of non-negative integers $x, y$ and $z$ that satisfy $2^x + 3^y = z^2$.

Problem 472
Evaluate $\displaystyle \int_0^{\infty} \frac{\ln(1 + x)\ln(1 + x^2)}{x^3} dx$

Last ones for a while now while I sit some tasty exams, gl on that integral, good job if someone gets it out before exams finish

Just was looking back a few pages and saw 472. Does that integral have a nice answer? I plugged it into Mathematica and the indefinite integral is horrific.

Edit: Surprisingly, I'm making some progress

Problem 485**

Prove that every open set $G$ on the real line is the union of a finite or countable system of pairwise disjoint open intervals, where we regard $(-\infty,x),(-\infty,\infty),(y,\infty)$ as open intervals.

Problem 485**

Prove that every open set $G$ on the real line is the union of a finite or countable system of pairwise disjoint open intervals, where we regard $(-\infty,x),(-\infty,\infty),(y,\infty)$ as open intervals.