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-   -   Approximate derivative of x! (https://www.mersenneforum.org/showthread.php?t=25528)

 othermath 2020-05-10 02:04

Approximate derivative of x!

Working experimentally, I found the approach

[TEX](x!)'=x!(\frac{x-2}{2(x-1)^2}+lnx)[/TEX]

For x>2 the approach improves as the value of x increases. I don't know if this is true for very large values ​​of x, (x € R).

Can you check it?

 Batalov 2020-05-10 02:38

[QUOTE=othermath;545004]I don't know if this is true for very large values ​​of x, (x € R).
[/QUOTE]
What is x! for x € R ?
Define it and then we can talk about derivatives.

 othermath 2020-05-10 03:17

[QUOTE=Batalov;545006]What is x! for x € R ?
Define it and then we can talk about derivatives.[/QUOTE]

I want to say that x is a real number.

 VBCurtis 2020-05-10 04:51

And what do you want to say the factorial of a real number is?

 retina 2020-05-10 05:16

Presumably n! = Γ(n + 1). For n >= 0 that would suffice, right? What did I miss?

 Batalov 2020-05-10 06:11

[QUOTE=retina;545011]Presumably n! = Γ(n + 1). For n >= 0 that would suffice, right? What did I miss?[/QUOTE]
You are missing that n! = Γ(n+1) + sin(πn). Or Γ(n+1) + sin(π(n+1)); take your pick... Wikipedia is helpful [URL="https://en.wikipedia.org/wiki/Gamma_function#/media/File:Gamma_plus_sin_pi_z.svg"]to demonstrate that both are valid analytic continuations of the factorials to the non-integers[/URL].
Do Γ(n+1) and Γ(n+1) + sin(πn) have the same derivatives?

 othermath 2020-05-10 06:16

[QUOTE=VBCurtis;545010]And what do you want to say the factorial of a real number is?[/QUOTE]
I assume ...

 othermath 2020-05-10 06:31

[QUOTE=Batalov;545012]You are missing that n! = Γ(n+1) + sin(πn). Or Γ(n+1) + sin(π(n+1)); take your pick... Wikipedia is helpful [URL="https://en.wikipedia.org/wiki/Gamma_function#/media/File:Gamma_plus_sin_pi_z.svg"]to demonstrate that both are valid analytic continuations of the factorials to the non-integers[/URL].
Do Γ(n+1) and Γ(n+1) + sin(πn) have the same derivatives?[/QUOTE]

Γ(n+1) requires the use of an integral. My relationship is simpler.

 othermath 2020-05-10 06:34

[QUOTE=retina;545011]Presumably n! = Γ(n + 1). For n >= 0 that would suffice, right? What did I miss?[/QUOTE]

That I answered Batalov.

 retina 2020-05-10 06:40

[QUOTE=Batalov;545012]You are missing that n! = Γ(n+1) + sin(πn). Or Γ(n+1) + sin(π(n+1)); take your pick... Wikipedia is helpful [URL="https://en.wikipedia.org/wiki/Gamma_function#/media/File:Gamma_plus_sin_pi_z.svg"]to demonstrate that both are valid analytic continuations of the factorials to the non-integers[/URL].
Do Γ(n+1) and Γ(n+1) + sin(πn) have the same derivatives?[/QUOTE]Thanks. Makes sense.

 Dr Sardonicus 2020-05-10 23:49

There is a fine little book (good luck finding a copy!) entitled [u]The Gamma Function[/u] by Emil Artin. In it he shows that the Gamma function is distinguished by being "log convex."

As [b]Retina[/b] has noted, x! = Γ(x+1) when x is a non-negative integer.

As to the derivative: There is a well-known asymptotic expansion [Stirling's asymptotic series] for ln(Γ(z)), z a complex variable. Taking the derivative term by term gives an asymptotic series for Γ'(z)/Γ(z).

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