1
$\begingroup$

Define $$f(t):=\int_0^1 \frac{\sqrt{x+t+\sqrt{t^2+2tx+1}}}{\sqrt{1-x^2}} dx $$ Can this integral be evaluated in terms of elliptic integrals? I ask because I have established the functional equation $$f(t)+f(-t)=2\pi \sqrt{2t}$$ And I hope to turn it into a potentially useful functional equation for elliptic integrals.

Thanks!

$\endgroup$
0
$\begingroup$

I am not sure that we could find any closed form for this integral and I suppose that you will need numerical integration to get $$f(t)=\int_0^1 \frac{\sqrt{x+t+\sqrt{t^2+2tx+1}}}{\sqrt{1-x^2}}\, dx$$ On the other hand, I am surprised by $$f(t)+f(-t)=2\pi \sqrt{2t}$$ Consider $t$ to be small; using Taylor expansions for the integrand and integrating, we have $$f(t) =2+t-\frac{t^2}{12}-\frac{t^3}{40}+\frac{5 t^4}{448}+O\left(t^5\right)$$ that is to say $$f(t)+f(-t)\approx 4-\frac{t^2}{6}+\frac{5 t^4}{224}+O\left(t^6\right)$$ which is "almost" exact for $-1 \leq t \leq 1$.

I produce below a table of numerical values $$\left( \begin{array}{cccc} t & f(t) & f(-t) & f(t)+f(-t) & 4-\frac{t^2}{6}+\frac{5 t^4}{224}\\ -3.00 & 0.444240 & 4.25406 & 4.69830 \\ -2.75 & 0.467638 & 4.10797 & 4.57561 \\ -2.50 & 0.495114 & 3.95676 & 4.45187 \\ -2.25 & 0.527996 & 3.79987 & 4.32787 \\ -2.00 & 0.568309 & 3.63669 & 4.20500 \\ -1.75 & 0.619341 & 3.46649 & 4.08583 \\ -1.50 & 0.686924 & 3.28841 & 3.97534 \\ -1.25 & 0.783061 & 3.10149 & 3.88456 \\ -1.00 & 0.945822 & 2.90465 & 3.85047 &3.85565\\ -0.75 & 1.21549 & 2.69676 & 3.91225 &3.91331\\ -0.50 & 1.48276 & 2.47686 & 3.95962 &3.95973\\ -0.25 & 1.74522 & 2.24445 & 3.98967 &3.98967\\ 0.00 & 2.00000 & 2.00000 & 4.00000 &4.00000\\ 0.25 & 2.24445 & 1.74522 & 3.98967 &3.98967\\ 0.50 & 2.47686 & 1.48276 & 3.95962 &3.95973\\ 0.75 & 2.69676 & 1.21549 & 3.91225 &3.91331\\ 1.00 & 2.90465 & 0.945822 & 3.85047& 3.85565\\ 1.25 & 3.10149 & 0.783061 & 3.88456 \\ 1.50 & 3.28841 & 0.686924 & 3.97534 \\ 1.75 & 3.46649 & 0.619341 & 4.08583 \\ 2.00 & 3.63669 & 0.568309 & 4.20500 \\ 2.25 & 3.79987 & 0.527996 & 4.32787 \\ 2.50 & 3.95676 & 0.495114 & 4.45187 \\ 2.75 & 4.10797 & 0.467638 & 4.57561 \\ 3.00 & 4.25406 & 0.44424 & 4.69830 \end{array} \right)$$

If $t$ is large, using Taylor again and integrating

$$f(t)=\frac{\pi \sqrt{t}}{\sqrt{2}}+\frac{1}{\sqrt{2t}}-\frac{1}{24 t^2\sqrt{2t}}+O\left(\frac{1}{t^{9/2}}\right)$$ identical to the expansion of the formula given by Mariusz Iwaniuk in his comment.

For $t=10$, the approximation leads to $7.24833$ while the exact value would be $7.24833$ (!!).

$\endgroup$
  • $\begingroup$ Closed form exist.CAS says: $\frac{\sqrt{2} \pi \sqrt{t \left(\sqrt{t^2+1}+t\right)}-2 \sqrt{2} \sqrt{t \left(\sqrt{t^2+1}+t\right)} \tan ^{-1}\left(\frac{1}{\sqrt{2} \sqrt{t \left(\sqrt{t^2+1}+t\right)}}\right)+4}{2 \sqrt{\sqrt{t^2+1}+t}}$ for $t>0$ $\endgroup$ – Mariusz Iwaniuk Jul 4 '18 at 10:14
  • $\begingroup$ @MariuszIwaniuk. This is a good and big surprise to me. Thanks for telling it ! Which CAS did you use ? Cheers. $\endgroup$ – Claude Leibovici Jul 4 '18 at 10:17
  • $\begingroup$ Mathematica 11.3. $\endgroup$ – Mariusz Iwaniuk Jul 4 '18 at 10:25
  • $\begingroup$ @MariuszIwaniuk. This does not seem to match the numerical results. Being amost blind, I may have made some typo's. Could you check, comparing to numerical integration ? Thanks for letting me know. In any manner, add an answer, please. $\endgroup$ – Claude Leibovici Jul 4 '18 at 10:27
  • $\begingroup$ Yours calculation are correct.It seems that: $f(t)+f(-t)\color{red}{\neq} 2 \pi \sqrt{2 t}$ $\endgroup$ – Mariusz Iwaniuk Jul 4 '18 at 11:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.