20200510, 02:04  #1 
May 2020
5_{8} Posts 
Approximate derivative of x!
Working experimentally, I found the approach
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? 
20200510, 02:38  #2 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2×3^{3}×13^{2} Posts 

20200510, 03:17  #3 
May 2020
5_{8} Posts 

20200510, 04:51  #4 
"Curtis"
Feb 2005
Riverside, CA
29·149 Posts 
And what do you want to say the factorial of a real number is?

20200510, 05:16  #5 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2^{2}×1,433 Posts 
Presumably n! = Γ(n + 1). For n >= 0 that would suffice, right? What did I miss?

20200510, 06:11  #6  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2·3^{3}·13^{2} Posts 
Quote:
Do Γ(n+1) and Γ(n+1) + sin(πn) have the same derivatives? 

20200510, 06:16  #7 
May 2020
5 Posts 

20200510, 06:31  #8  
May 2020
5_{10} Posts 
Quote:


20200510, 06:34  #9 
May 2020
5 Posts 

20200510, 06:40  #10  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
5732_{10} Posts 
Quote:


20200510, 23:49  #11 
Feb 2017
Nowhere
23·151 Posts 
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 nonnegative integer. As to the derivative: There is a wellknown 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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