Message Boards Message Boards

8
|
4681 Views
|
16 Replies
|
34 Total Likes
View groups...
Share
Share this post:

[CALL] for challenging univariate symbolic integrals

enter image description here

This is a call for univariate indefinite or definite exact integrals that cannot immediately computed in Mathematica and Wolfram Language. Submitted integrals will be collected for potential new event where they will be discussed and solved. You may refer to our previous event, which is showcased on the Wolfram Community post: https://community.wolfram.com/groups/-/m/t/2821053.
Do you have an integral that cannot be computed directly using Integrate ? Does the following apply to this integral?

Submission rules DOs

  • Univariate (terms in a single variable)
  • Exact (use only symbols, integers, or exact fractions).
  • Definite integrals (1, 2)
  • Indefinite Integrals (1, 2)
  • Keep comments directly related to the topics.

Submission rules DON'Ts

  • Don't submit those that Mathematica / Wolfram Language can do directly using Integrate[ ].
  • Don't submit numerical integrals (NIntegrate[ ])
  • Don't submit decimal approximate numbers in Integrate[ ].
  • Don't submit images, only copyable code.

If you have such an integral - please put it in a comment below!

Examples of submission formats

You can add a notebook with your integral:

You can post a formatted code:

Integrate[Gamma[s] z^-s, s]
POSTED BY: Oleg Marichev
16 Replies

I tried another very popular book with random integrals with Mathematica that can't solve:

Attachments:
POSTED BY: Mariusz Iwaniuk

I opened another book about LaplaceTransform and tried a few examples:

      LaplaceTransform[BesselJ[v, a x] BesselI[v, b x], x, s, 
       Assumptions -> {Re[v] > 0, a > 0, b > 0}](*Can't Solve !*)

      LaplaceTransform[(1 - Exp[-x])^v* Erfc[ a (1 - Exp[-x])], x, s, 
       Assumptions -> {Re[v] > -1, a > 0}](*Can't Solve !*)

      LaplaceTransform[(1 - Exp[-x])^\[Mu]* Gamma[v, a*(1 - Exp[-x])], x, s,
        Assumptions -> {Re[\[Mu]] > -1, Re[v] > 0, a > 0}](*Can't Solve !*)

      LaplaceTransform[(x + a)^n*LegendreP[n, (x - a)/(x + a)], x, s, 
        Assumptions -> {Re[n] > 0, a > 0}](*Can't Solve !*)
POSTED BY: Mariusz Iwaniuk

I opened a book about MellinTransform and tried a few examples:

    MellinTransform[(Sqrt[x^2 + a^2] + a)^\[Rho]/Sqrt[x^2 + a^2]*
     BesselJ[v, b*x], x, s, Method -> "Integrate", 
    Assumptions -> {b > 0, 
      Re[a] > 0, -Re[v] < Re[s] < 5/2 - Re[\[Rho]]}](*Can't Solve !*)

   MellinTransform[
    Exp[-\[Sigma] Sqrt[x] - \[Omega]*x] Hypergeometric1F1[a, 
      b, \[Omega]*x], x, s, Method -> "Integrate", 
    Assumptions -> {a > 0, b > 0, Re[\[Sigma]] > 0, 
      Re[s] > 0}](*Can't Solve !*)

   MellinTransform[
    1/(x + \[Sigma])^\[Rho] Hypergeometric2F1[a, b, 
      c, -\[Omega]*x], x, s, Method -> "Integrate", 
    Assumptions -> {Re[\[Rho]] > 0, Re[\[Sigma]] > 0, 
      Re[s] > 0}](*Can't Solve !*)
POSTED BY: Mariusz Iwaniuk

I would be very interested in seeing a solution for this definite parameter integral:

Integrate[((1 - x)^r (-2 x + (-2 + x) Log[1 - x]))/(2 x Log[1 - x]^2), {x, 0, 1}, Assumptions -> r >= 0]
POSTED BY: Henrik Schachner

Looks like: $$\int_0^1 \frac{(1-x)^r (-2 x+(-2+x) \log (1-x))}{2 x \log ^2(1-x)} \, dx=1+r-r \log (1+r)-\frac{1}{2} \log (2 \pi (1+r))+\text{log$\Gamma $}(1+r)$$

 Integrate[((1 - x)^r (-2 x + (-2 + x) Log[1 - x]))/(2 x )*t (1 - x)^
   t, {t, 0, Infinity}, {x, 0, 1}, Assumptions -> {r >= 0, t >= 0}]

 (*1 + r - r Log[1 + r] - 1/2 Log[2 \[Pi] (1 + r)] + LogGamma[1 + r]*)

Regards M.I

POSTED BY: Mariusz Iwaniuk

Nice trick ... :)

It should be nice to generalize Integrate function to be able to solve this directly:

Integrate[((1 - x)^r (-2 x + (-2 + x) Log[1 - x]))/(2 x Log[1 - x]^2), {x, 0, 1}, Assumptions -> r >= 0]

using (in this specific case) the following identity:

1/Log[1 - x]^2 = Integrate[t (1 - x)^t, {t, 0, Infinity}] if Re[Log[1 - x]] < 0

like

Integrate[((1 - x)^r (-2 x + (-2 + x) Log[1 - x]))/(2 x)*
  t (1 - x)^t, {t, 0, Infinity}, {x, 0, 1}, Assumptions -> r >= 0]

What a nice trick, indeed, I am impressed! Many thanks, Mariusz!

POSTED BY: Henrik Schachner

I have a site of Table of Define integrals but you have to go through the whole list and see if Integrate can solve.

Attachments:
POSTED BY: Mariusz Iwaniuk

Another list for define integrals with attached files.

POSTED BY: Mariusz Iwaniuk

Another list for define integrals with attached files.

Examples from here.

POSTED BY: Mariusz Iwaniuk

POSTED BY: Frank Iannarilli

I have a site of Table of indefine integrals but you have to go through the whole list and see if Integrate can solve. Another site which can be very helpful.

My list attached below

Attachments:
POSTED BY: Mariusz Iwaniuk

2 another indefine integrals:

  Integrate[Exp[-a Sqrt[x]]*Exp[-a*Sqrt[ x - b]], x](* 1/2 b (E^(-a Sqrt[b]
        x) (1/x - (2 + 2 a Sqrt[b] x + a^2 b x^2)/(a^3 b^(3/2))) + 
     a Sqrt[b] ExpIntegralEi[-a Sqrt[b] x])*)

  Integrate[Gamma[a , x]^2/x^2, x](* -((2 E^-x x^(-1 + a) Gamma[a, x])/(-1 + a)) + Gamma[a, x]^2/(-1 + a) +
    Gamma[a, x]^2/((-1 + a) x) - (a Gamma[a, x]^2)/((-1 + a) x) + (
   2^(2 - 2 a) Gamma[-1 + 2 a, 2 x])/(-1 + a)*)
POSTED BY: Mariusz Iwaniuk

Hi Oleg, since a long time I don't come to terms with all the orthogonality relations of polynomial families.

Just after the invention of the internet I found your books at Oxfords Blackwells,, ordered the four volumes and paid about 1000 GBP :-( The next days I found Mathenatica 2.0 in Hamburg for 50€ :-).

Here is my attempt with Jacobi polynomials (Gradshtey/Ryzhik 7.391) (Notebook attached)

Regards Roland

Attachments:
POSTED BY: Roland Franzius

Here is one:

POSTED BY: Ahmed Elbanna
Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Reply Preview
Attachments
Remove
or Discard

Group Abstract Group Abstract