Timeline for Is there a specific named function that is the inverse of $x+x^a$ for $x$ real?
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17 events
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| Nov 7, 2023 at 2:23 | comment | added | Jorge Zuniga | @TymaGaidash. I am afraid that sum has no a closed form. You should check if $|y^{\alpha-1}|<|(\alpha-1)^{\alpha-1}\alpha^{-\alpha}|$ holds in this case. | |
| Nov 6, 2023 at 17:20 | comment | added | Тyma Gaidash | @JorgeZuniga There does not seem to be a formula for $\Gamma(a+zn)\to\Gamma(b+xn),z\in\Bbb C,x\in\Bbb R$ for the integral. It would be ideal if one could write, for example: $u=-1-e^{(\sqrt3+i)\pi k}u^{(-1)^{-\frac13}}\implies u_k= -\sum\limits_{n=0}^\infty\frac{e^{\sqrt[6]{-1}(2k+1)\pi n}}{n!}\frac{\Gamma\big(1+(-1)^{-\frac13}n\big)}{\Gamma\big(2+(-1)^\frac43n\big)}= -\,_1\Psi_1\left(_{2,(-1)^\frac43}^{1,(-1)^{-\frac13}}; e^{\sqrt[6]{-1}(2k+1)\pi }\right)$. Are you sure sums, or integrals, like these do not have closed forms? | |
| Nov 6, 2023 at 16:19 | comment | added | Jorge Zuniga | @TymaGaidash Classic definitions of Fox Wright and Fox H need those parameters positive reals. You can extend them to the negative reals using the reflexion formula for Gammas. For complex values it is better to see what happens analyzing the Mellin Barnes Integral definition for both functions, but I have never seen if this case is possible. | |
| Nov 5, 2023 at 15:11 | comment | added | Тyma Gaidash | @JorgeZuniga The series expansion works for complex $a$ also, but Fox H cannot have complex $\alpha_j,\beta_j$. Do you know if one can extend Fox H like so or if the Fox Wright function can have complex $A_j, B_j$? The Fox Wright function is also not exactly clear if it is a standardized function, so it is hard to see if for complex $a$, either function gives a closed form for $x^a+x=t$. | |
| Jan 2, 2022 at 1:16 | comment | added | Jorge Zuniga | @Mariusz_Iwaniuk. Since FoxH is a new function, I think this is a question for Mathematica & Wolfram Language StackExchange Site. I will ask to check it. These expressions come from Fox-Wright function. There are slightly different formulae using FoxH function for all roots of trinomial equations in Wolfram's site reference.wolfram.com/language/ref/FoxH.html (Applications Section) | |
| Dec 31, 2021 at 14:42 | comment | added | Mariusz Iwaniuk |
I tried to find solution to equation: $x^2+x=1$ in Mathematica code:N[y*FoxH[{{{0, \[Alpha]}}, {{}}}, {{{0, 1}}, {{-1, \[Alpha] - 1}}}, y^(\[Alpha] \[Minus] 1)] /. \[Alpha] -> 2 /. y -> 1] ,but I can't get numeric value?
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| Nov 25, 2021 at 0:03 | history | edited | Jorge Zuniga | CC BY-SA 4.0 |
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| Nov 14, 2021 at 21:58 | history | edited | Jorge Zuniga | CC BY-SA 4.0 |
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| Nov 13, 2021 at 3:35 | history | edited | Jorge Zuniga | CC BY-SA 4.0 |
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| Nov 12, 2021 at 2:57 | comment | added | Jorge Zuniga | @j-ham, An introduction to (Extended) Generalized Hypergeometric Functions (MeijerG, Fox-Wright and Fox-H) can be found in the link Fox-H function above. Several reference books are found at the bottom of this document. Fox-Wright function, as a special case of Fox-H function, can be computed through Wolfram's Mathematica (version 12.3). | |
| Nov 10, 2021 at 14:56 | comment | added | J.Ham | This is very helpful. Now can one use the Fox-Wright to solve y=x^𝛼-x? I am finding it difficult to hit on a reference book here as Fox-Wright doesn't appear in the DLMF. | |
| Nov 10, 2021 at 14:20 | vote | accept | J.Ham | ||
| Nov 6, 2021 at 0:57 | history | edited | Jorge Zuniga | CC BY-SA 4.0 |
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| Nov 5, 2021 at 5:11 | history | edited | Jorge Zuniga | CC BY-SA 4.0 |
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| Nov 5, 2021 at 4:16 | history | edited | Jorge Zuniga | CC BY-SA 4.0 |
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| Nov 5, 2021 at 0:31 | history | edited | Jorge Zuniga | CC BY-SA 4.0 |
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| Nov 4, 2021 at 22:51 | history | answered | Jorge Zuniga | CC BY-SA 4.0 |