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It seems that there are real integrals that are immune to all real methods of integration and one has to apply the residue theorem and contour integration. Here is my collection

$$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}\tag {1}$$

$$\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{(e^{x}-x)^{2}+{\pi}^{2}}=\frac{1}{1+W(1)}=\frac{1}{1+\Omega}\tag {2}$$

$$\int^{\pi/2}_{0}\cos(xt)\cos^y(t)\,\mathrm{d}t=\frac{\pi \Gamma(x+1)}{2^{y+1}\Gamma\left(\frac{x+y+2}{2}\right)\Gamma\left(\frac{2-x+y}{2}\right)}\tag{3}$$

$$\int_0^{1}\arctan\left(\frac{\mathrm{arctanh}\ x-\arctan{x}}{\pi+\mathrm{arctanh}\ x-\arctan{x}}\right)\frac{\mathrm{d}x}{x}=\frac{\pi}{8}\log\frac{\pi^2}{8} \tag {4}$$

$$\int_{-\infty}^{\infty} \frac{\mathrm{d}x}{(e^x+x+1)^2+\pi^2}=\frac{2}{3}\tag {5}$$

Are there any other examples ?

References

  1. https://artofproblemsolving.com/community/c7h501365p2817263
  2. Interesting integral related to the Omega Constant/Lambert W Function
  3. http://advancedintegrals.com/2017/04/integrating-a-function-around-three-branches-using-a-semi-circle-contour/
  4. https://arxiv.org/pdf/1402.3830.pdf
  5. Integral: $\int_{-\infty}^{\infty} \frac{dx}{(e^x+x+1)^2+\pi^2}$
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Zaid Alyafeai
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  • but these integrals are not immune to numerical approximations, right (as Simpson rule or Euler-Maclaurin sum formula)? I mean, if an integral doesnt have a primitive expressible in elementary functions we need numerical integration methods, except for some special cases. – Masacroso Apr 24 '17 at 07:25
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    @Masacroso, of course but you won't get that nice closed form solution like the first one using approximations. My question isn't about integrals that can't be expresses using elementary function's rather those where contour integrations seem superior. – Zaid Alyafeai Apr 24 '17 at 07:29
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    In the third integral, if m,n>0 are integers, no need contour integration imho – FDP Apr 24 '17 at 08:34
  • @FDP, They are real. Hence the use of the Gamma function. – Zaid Alyafeai Apr 24 '17 at 09:25
  • I changed $m,n $ to $x,y $ so they are not confused to be integers. – Zaid Alyafeai Apr 24 '17 at 10:50
  • For your last one, $$-\frac{\cos x}{2(2\cos x+e^{\sqrt{3}x})}$$ is a primitive function, so that one is trivial. Also, your question would be better if you gave the source and some argument for your examples. – mickep Apr 24 '17 at 11:16
  • @mickep, you are right. I will add the references , the complex functions and contours later. – Zaid Alyafeai Apr 24 '17 at 11:28
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    https://math.stackexchange.com/questions/1055468/integral-int-infty-infty-fracdxexx12-pi2/1055598#1055598 – Ron Gordon Apr 24 '17 at 11:52
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    Real and imaginary parts of Fourier transforms most of the time require residue theorem. To my knowledge, $\int_\Bbb R \frac{\cos x}{1+x^2}dx$ can't be found by "real methods". – TZakrevskiy Apr 24 '17 at 11:54
  • @RonGordon, thanks added. – Zaid Alyafeai Apr 24 '17 at 12:17
  • @TZakrevskiy, I always thought the same. Not sure if that's true. – Zaid Alyafeai Apr 24 '17 at 12:19
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    @ZaidAlyafeai i'm quite confident that $(3)$ can be done without contour integrals/residue theorem. Furthermore https://math.stackexchange.com/questions/253910/the-integral-that-stumped-feynman might ne intersting for you – tired Apr 24 '17 at 17:32
  • @ZaidAlyafeai How does one know a definite integral, which can be evaluated using complex analysis, cannot be evaluated using real analysis only? – Mark Viola Apr 24 '17 at 17:32
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    @TZakrevskiy [See This Answer](https://math.stackexchange.com/questions/2236261/find-the-fourier-cosine-transform-of-the-function-defined-by-displaystyle-fx/2236287#2236287) in which I presented a real analysis method for evaluating the Fourier Cosine Transform $\int_{-\infty}^\infty \frac{\cos(\omega x)}{1+x^2}\,dx$. – Mark Viola Apr 24 '17 at 17:41
  • @tired Of course, Feynman's failure doesn't imply that it is impossible. In fact, since the residue theorem is effectively a consequence of Green's Theorem in the plane, contour integration can be translated into an equivalent problem of analysis in the real plane. Does that make sense? – Mark Viola Apr 24 '17 at 17:45
  • @tired, I feel the same towards (3) but never seen a solution that involves only real methods. Actually I posted that question around two weeks ago nobody responded https://math.stackexchange.com/questions/2227196/real-methods-for-the-evaluating-int-pi-2-0-cosnt-cosmt-dt – Zaid Alyafeai Apr 24 '17 at 18:27
  • @Dr.MV, I don't know. But the easiness of solving the integral using contour integration seems to deviate people from trying real analysis methods. Something like (1) seems to take contour integration to another level with the complex parametrization it involves . – Zaid Alyafeai Apr 24 '17 at 18:34
  • Zaid, I'd love to see references or links to where these integrals have been evaluated. Do you have them? -Mark – Mark Viola Apr 24 '17 at 19:00
  • @Dr.MV, I will add everything later. I am quite ill and only using my phone. – Zaid Alyafeai Apr 24 '17 at 19:30
  • @ZaidAlyafeai I hope you feel better soon. – Mark Viola Apr 24 '17 at 19:47
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    Probably a piece of cake for Ramanujan to derive these results, but note that he didn't know about complex analysis until quite late in his short life. – Count Iblis Apr 24 '17 at 20:12
  • @Dr.MV some quick thinking makes your argument plausible, but i have never seen an actual prove for this equivalence. Can you provide one? the problem seems to to translate the concept of holomorpic functions appropriately – tired Apr 24 '17 at 21:27
  • @tired It's fairly straightforward, but a bit too long for a comment. I'd post a solution if someone posted the question. -Mark – Mark Viola Apr 24 '17 at 22:01
  • @Dr.MV I included the references for proofs of all the questions. Most of the links are external unfortunately. – Zaid Alyafeai Apr 25 '17 at 01:43
  • https://math.stackexchange.com/questions/1819837/integral-int-infty-infty-frac-gammax-sin-pi-x-gamma-leftxa-rig?rq=1 This has no real solution.... – Jack Tiger Lam Apr 25 '17 at 04:28
  • https://math.stackexchange.com/questions/331148/interesting-integral-int-infty-infty-fracei-nx-gamma-alphax-g?rq=1 – Jack Tiger Lam Apr 25 '17 at 04:32
  • @Dr.MV that's quite nice, thank you. – TZakrevskiy Apr 25 '17 at 08:34
  • Do you mean log (base 10) or ln ? – Jasper Aug 30 '18 at 04:54

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