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Old 2020-05-10, 02:04   #1
othermath
 
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Default Approximate derivative of x!

Working experimentally, I found the approach

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

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?
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Old 2020-05-10, 02:38   #2
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Quote:
Originally Posted by othermath View Post
I don't know if this is true for very large values ​​of x, (x € R).
What is x! for x € R ?
Define it and then we can talk about derivatives.
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Old 2020-05-10, 03:17   #3
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Originally Posted by Batalov View Post
What is x! for x € R ?
Define it and then we can talk about derivatives.
I want to say that x is a real number.
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Old 2020-05-10, 04:51   #4
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And what do you want to say the factorial of a real number is?
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Old 2020-05-10, 05:16   #5
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Presumably n! = Γ(n + 1). For n >= 0 that would suffice, right? What did I miss?
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Old 2020-05-10, 06:11   #6
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Quote:
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Presumably n! = Γ(n + 1). For n >= 0 that would suffice, right? What did I miss?
You are missing that n! = Γ(n+1) + sin(πn). Or Γ(n+1) + sin(π(n+1)); take your pick... Wikipedia is helpful to demonstrate that both are valid analytic continuations of the factorials to the non-integers.
Do Γ(n+1) and Γ(n+1) + sin(πn) have the same derivatives?
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Old 2020-05-10, 06:16   #7
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Quote:
Originally Posted by VBCurtis View Post
And what do you want to say the factorial of a real number is?
I assume ...
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Old 2020-05-10, 06:31   #8
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Quote:
Originally Posted by Batalov View Post
You are missing that n! = Γ(n+1) + sin(πn). Or Γ(n+1) + sin(π(n+1)); take your pick... Wikipedia is helpful to demonstrate that both are valid analytic continuations of the factorials to the non-integers.
Do Γ(n+1) and Γ(n+1) + sin(πn) have the same derivatives?
Γ(n+1) requires the use of an integral. My relationship is simpler.
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Old 2020-05-10, 06:34   #9
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Quote:
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Presumably n! = Γ(n + 1). For n >= 0 that would suffice, right? What did I miss?
That I answered Batalov.
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Old 2020-05-10, 06:40   #10
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Quote:
Originally Posted by Batalov View Post
You are missing that n! = Γ(n+1) + sin(πn). Or Γ(n+1) + sin(π(n+1)); take your pick... Wikipedia is helpful to demonstrate that both are valid analytic continuations of the factorials to the non-integers.
Do Γ(n+1) and Γ(n+1) + sin(πn) have the same derivatives?
Thanks. Makes sense.
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Old 2020-05-10, 23:49   #11
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There is a fine little book (good luck finding a copy!) entitled The Gamma Function by Emil Artin. In it he shows that the Gamma function is distinguished by being "log convex."

As Retina 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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