So I tried evaluating it and ended up with π, but the answer is π/2 and can't find where I'm wrong
#Integration
void jackal
abstract bluffBOT
So I tried evaluating it and ended up with π, but the answer is π/2 and can't fi...
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void jackal
This was what I did
brave steeple
Is it sin(log(2)(sin(x)) or sin(log(2 sin(x)))
void jackal
Is it sin(log(2)(sin(x)) or sin(log(2 sin(x)))
The first
fluid snow
So I tried evaluating it and ended up with π, but the answer is π/2 and can't fi...
This problem looks awfully familiar, but anyways, the numerator should be $\Im 2^{e^{ix}}$
weak daggerBOT
Accelerator
fluid snow
And then from there, I would imagine a contour with the shape of 1/4th of a donut in the first quadrant would be worth trying out
Or possibly using a series could work too, assuming the interval of integration lies inside the circle of convergence of the series so that one could swap the sum and integral, but that doesn't seem very obvious to me
void jackal
This problem looks awfully familiar, but anyways, the numerator should be $\Im 2...
Indeed that is what I did do
I didn't follow up with contour integration though, I used a series expansion
I just can't seem to find where exactly the error is
brave steeple
How did you verify the answer is pi?
fluid snow
Indeed that is what I did do
Wait, so when you wrote $e^{\ln2e^{ix}}$, did you mean $e^{\ln(2)e^{ix}}$
weak daggerBOT
Accelerator (Cody)
void jackal
Wait, so when you wrote $e^{\ln2e^{ix}}$, did you mean $e^{\ln(2)e^{ix}}$
Yes I did
Sorry about that
void jackal
How did you verify the answer is pi?
It's π/2, I checked on Desmos
What I am getting as an answer is π
fluid snow
I would be careful about determining whether or not you can pull the imaginary part put of the integral because $\frac{2^{e^{ix}}}{x}$ has a simple pole at $x=0$ which makes $\int_0^{\infty} \frac{2^{e^{ix}}}{x}dx$ by itself non-improperly integrable regardless of the $\Im$ outside
weak daggerBOT
Accelerator (Cody)
void jackal
Interesting
$\lim_{\epsilon \to 0^+} \int_{\epsilon}^{\infty} \Im \frac{2^{\exp ix}}{x} \dd x$
weak daggerBOT
NEON
void jackal
Can I pull it out now then
fluid snow
Can I pull it out now then
Yeah as long as the Im part is to the right of the limit because the function is integrable over any domain containing a neighborhood of epsilon
But there's another completely separate problem aside from the Im stuff
brave steeple
I hate to be that guy but have you tried contour integrating it over some finite semicircle contour and sending it's lim to inf
void jackal
I have considered that but I'm not really all that comfortable with contour integration yet
Only done a few problems with it
The residues are easy enough to calculate I suppose
brave steeple
And are you sure Desmos is right cuz i tried checking this with mathematica and it keeps timing out for me?
void jackal
Indeed even Wolfram times out
brave steeple
I didn't even know demos had a cas inbuilt
void jackal
Lol
brave steeple
Numerical int evaluated something like -53 some gibberish
fluid snow
**NEON**
Did you do a substitution $x = \frac{u}{k}$? It wouldn't work if $k=0$ and your sum goes from $k=0$ to $\infty$
weak daggerBOT
Accelerator (Cody)
void jackal
I just picked 1000 for infinity and it gave me a value that was unmistakably π/2
void jackal
Did you do a substitution $x = \frac{u}{k}$? It wouldn't work if $k=0$ and your ...
holy shite
void jackal
oooo
I think that was it cody
$\frac{\pi}{2} \cdot \sum_{k \geq 1} \frac{\ln^k 2}{k!}$
weak daggerBOT
NEON
void jackal
Now I gotta do + π/2 - π/2 in order to get e^(ln 2)
brave steeple
It was a fencepost error?
void jackal
$\frac{\pi}{2} \sum_{k \geq 0} \frac{\ln^k 2}{k!} - \frac{\pi}{2} = \frac{\pi}{2}$
weak daggerBOT
NEON
fluid snow
I think you got it
void jackal
Can't believe I fell victim to such a classic blunder
fluid snow
Someone made a similar mistake here if you can catch it https://math.stackexchange.com/questions/4739613/conjecture-about-the-value-of-int-infty-infty-frace-cosx-sin-sin
void jackal
thanks a lot mate
brave steeple
Someone made a similar mistake here if you can catch it https://math.stackexchan...
Oh wow this seems someone copying homework but changing a minor detail lol
void jackal
Who me?
fluid snow
So I'm pretty sure that would be a common mistake because I had a hard time finding the error too
void jackal
Mental note that sin(0) = 0 from next time
void jackal
Ah fair
fluid snow
I hate to be that guy but have you tried contour integrating it over some finite...
I've tried that with a similar problem before
void jackal
I haven't derped around MSE too much, I could find some interesting integrals there I bet
I'm a bit of an integral addict haha
brave steeple
I've tried that with a similar problem before
Yea it's always the last ditch hail mary tbh you never wanna be forced to resort to it
void jackal
Do I somehow close this or?
fluid snow
Yea it's always the last ditch hail mary tbh you never wanna be forced to resort...
Nahhhhh that's my first option lmao
fluid snow
Do I somehow close this or?
Idk I'm new to this Discord lol