Share interesting integrals ITT.[math]\int_{a}^{b} (x-a)^n(x-b)^n \mathrm{d} x[/math]
>>16505839Isn’t this just a variant of Euler’s beta function?
>>16505840I checked out and it indeed seems to be.
>>16505839The logarithmic integral is pretty cool, especially for the prime number theorem[eqn]\text{li}(x)=\int_0^x \frac{1}{\ln(t)}\text{d}t\\\\ \lim_{x \to \infty } \frac{\pi(x)}{\text{li}(x)}=1\\ \text{where } \pi(x) \text{ is the prime number counting function:}\\ \pi(x): \text{number of primes less or equal than x}[/eqn]
[math]\mathfrak{B}=\{f\in C(\mathbb{R}) : \int\int\int f(x)d^3x< \frac{e^{i\pi}}{11.999\dots}\} [/math]The Barnett integrable functions have been a source of mystery for many years, I suspect there are profound theorems hidden within this set, which would change the world forever.
>>16505880You could have greentexted Perelman’s quote, you know
>>>>16506100crosspoast im publishing on 4chan
>>16492523
>>16505839The infamous Cleo's integral...The integral doesn't have an official name and isonly one of the 39 difficult integrals they havesolved on MSE during that time (2013-2015).The integral gained its fame from the unique and unusual solution that arose when solved...and fame for Cleo who presented the answer as iswithout any explanation. Based, if you asked me.
>>16505839This one is from >>16499324
>>16505839Kings Rule
>>16505881maybe he is perelman
>>16505839Are there any almost continuous functions that are Lebesgue integrable but not absolute (improper) Riemann integrable or vice versa?
>>16508576>and vice versanever happens because Riemann measure is a type of Lebesgue measure
>>16505839i dont even solve math by hand i just conceptualize it and its non linear patterns then find what to optimize for thats it if i really need to i use symbolab im a half ass quant
>>16508578What are you talking about?
>>16508756Riemann and Lebesgue measures
>>16508759What is a Riemann measure?
>>16508761A measure used in Riemann integrals?
>>16507265Undisputed goat
>>16508763Riemann integrals do not use measure.
>>16505839Riemann-Stieltjes integrals in general.[math]\int_a^b f(x) dg(x)[/math]
>>16509070Every integral does. Read a book.
>>16509082Okay, why does MCT fail for Riemann integral if it uses a measure?
>>16505839An integral inequality from >>16509888
>>16507265Doesn't look too bad.Choose the sqrt branch cut to be [-1,1].Write 2I as the cc integral around [-1,1] where the value just below the branch cut corresponds to branch intended in the integral.Deform the contour so that it becomes two clockwise integrals around the log branch cuts: i/2 + [-1/2, 1/2], -i/2 + [-1/2, 1/2]Use formula:[math]\int_a^b f(x)dx={1 \over 2 \pi i}\oint_{[a,b]}f(z)log({z-a \over z-b})dz[/math]
>>16511332that’s really neat. The argument of log looks a lot like a Moebius transformation. Is there something going on symmetry-wise?
>>16511332>>16507265Now, how about this integral of a hypergeometric function that Cleo found a closed form for in 2014?
>>16511410Idk if there is much to do with mobius.There are a few ways to understand, though.1) If you differentiate wrt a or b, you get the usual residue formula. Since you end up with opposite signs for f(a) and f(b), it sort of explains why one is the numerator and the other is denominator. You can also just add the integral for [a,b] and [b,c] and things telescope the way they should.2) The log is just a result of doing riemann sums via cauchy residue.[math]lim_{\Delta \rightarrow 0} {1 \over 2 \pi i}\oint_{[a,b]}f(z)\sum\limits_{n=0}^{{b-a \over \Delta}}{\Delta \over z-(a+n\Delta)}dz =\\ {1 \over 2 \pi i}\oint_{[a,b]}f(z)(\int_a^b{dw \over z-w})dz.[/math]
>>16505839this is interesting
>>16505839This one right here
>>16510467This is another thread of mine. I found it on an old hard math problems website.
>>16508461a classic!
>>16511696>The log is just a result of doing riemann sums via cauchy residue.Interesting. This looks like some kind of "discretization" of the interval [a,b] into a countable set of equidistant poles. But from this argumen f(z) must be entire?
>>16515210f just needs to be holo on the interior of the contour that circles [a,b].
>>16511411Greetings anonWhich is the function inside?
>>16517554The function inside is [math] {{1}\over{x^{1/2}}}\cdot \left[{}_3F_2\left(r,s,t;u,v|-x\right) \right]^2 [/math] where r,s,t are the top 3 fractions and u,v are the bottom 2 fractions.
>>16516782But the original function has a pole at the origin…
>>16518042In this special case there is no pole really there because the log zero cancels the 1/x.Since this is true, you can deform your initial integral path off the real line.Then once you have the two isolated loops you can apply the result.I will draw the deformations.
>>16507265based
[math]\int_{0}^{\infty} \frac{\sin(x)}{x} \, dx = \frac{\pi}{2}[/math]
[math]\int_{0}^{\infty} x^{k-1} e^{-x} \, dx = \Gamma(k)[/math]
What’s a good book to learn integration properly?
>>16505839
>>16505868that is cool
ends and coendsidk what they do
i fucking hate integrals so much, i cheated through all the classes fuck
>>16505839Here's another
[eqn] \int\limits_{0}^{x} \frac{log(1-t)}{t}dt = -Li_{2}(x) [/eqn].Cool particular case is[eqn] \int\limits_{0}^{1} \frac{log(1-t)}{t}dt = -\frac{pi^2}{6}[/eqn] (Basel Problem).
>>16531565Basel problem? More like based problem
>>16505839>No one posted the best integral in an entire monthIt's over for /sci/[math] \int_\mathbb{R} e^{ikx} \,dx = 2 \pi \delta(k) [/math]
>>16533264>Thinking of distributional identities as actual integralskys physics nigger
>>16533287You shouldn't suck so many cocks with that foul mouth of yours
>>16522098how do you solve that one?
>>16533287Dirac delta is just the Kronecker delta of continuum.
