oleg1
March 13, 2023, 3:01pm
1
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: [R&DL] Wolfram R&D LIVE: Ask Integration Questions to Oleg Marichev - Staff Picks - WOLFRAM COMMUNITY . Integrate ? Does the following apply to this integral?
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.
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!
You can add a notebook with your integral:
You can post a formatted code:
Integrate[Gamma[s] z^-s, s]
10 Likes
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
Here is my attempt with Jacobi polynomials (Gradshtey/Ryzhik 7.391) (Notebook attached)
Regards Roland
Integral submission.nb (83.2 KB)
2 Likes
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
111432973.nb (195 KB)
2 Likes
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)*)
1 Like
I have a site of Table of Define integrals but you have to go through the whole list and see if Integrate can solve.
Define integrals.nb (77.1 KB)
3 Likes
Another list for define integrals with attached files.
Table Całek Oznaczonych 1.nb (99.9 KB)Table Całek Oznaczonych 2.nb (97.8 KB)Table Całek Oznaczonych 3.nb (47.6 KB)
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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]
2 Likes
Another list for define integrals with attached files.
Examples from here.
Tintegrate integrals.nb (17.8 KB)
1 Like
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
4 Likes
What a nice trick, indeed, I am impressed! Many thanks, Mariusz!
1 Like
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 !*)
1 Like
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 !*)
1 Like
I tried another very popular book with random integrals with Mathematica that can’t solve:
Define Integrals.nb (14.1 KB)
1 Like
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]
1 Like
The next days I found Mathenatica 2.0 in Hamburg for 50€ :-).