Timeline for How are the Legendre Polynomials of second kind for negative degrees defined?
Current License: CC BY-SA 4.0
12 events
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| Jul 11, 2023 at 13:19 | vote | accept | BenjaminBluemchen | ||
| Jul 10, 2023 at 13:37 | comment | added | Carlo Beenakker |
the third formula is returned by FullSimplify[Hypergeometric2F1Regularized[a, b, 0, x]
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| Jul 10, 2023 at 13:21 | comment | added | BenjaminBluemchen | I see, I was running the same line in Mathematica and received the same output. But I still have the same issue in Matlab. Do you have a reference for the third formula? The documentation of Mathematica says $\bar{F}_1(a,b;c;x)=F_1(a,b;c;x)/\Gamma(c)$ which gives me a different but also wrong result | |
| Jul 10, 2023 at 12:40 | comment | added | Carlo Beenakker |
I compared the two expressions with Mathematica, and they are identical: FullSimplify[Gamma[1/2]*Gamma[nu + 1]* Hypergeometric2F1[nu/2 + 1, nu/2 + 1/2, nu + 3/2, 1/z^2]/(2^(nu + 1)*z^(nu + 1)*Gamma[nu + 3/2])]
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| Jul 10, 2023 at 12:29 | comment | added | BenjaminBluemchen | Thank you for adding more explainations. However I wasn't able to verify your transformation from the first to the second formula for $Q^0_\nu(z)$ by implementation, is it possible that there is a typo? | |
| Jul 10, 2023 at 11:05 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jul 10, 2023 at 10:36 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jul 10, 2023 at 10:30 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jul 10, 2023 at 10:13 | comment | added | Carlo Beenakker | I don't know how Mathematica implements LegendreQ, but there is no need to go through Mathematica, you can just take the limit of Gradshteyn's expression. | |
| Jul 10, 2023 at 10:11 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jul 10, 2023 at 10:06 | comment | added | BenjaminBluemchen | Thank you very much. Do you know where I can find more explanatory material on the implementation of Mathematicas LegendreQ. I would like to understand where the real and imaginary part is coming from. | |
| Jul 10, 2023 at 10:04 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |