/sci/ - Science & Math

[Anonymous No.8469665]

Why does the gamma function equals (n-1)! for the integers and not n!?

[Anonymous No.8469752]

>>8469665
Because then it would not be the gamma function.

Defining a new function with your characteristics is very easy given the gamma function.

[Anonymous No.8469784]

>>8469665
The simplest answer is there's really no good reason for it at all. The function you're describing does exist, it's called the Pi function and it was used for a while before the modern Gamma was defined.

There's really no solid justification known for why Legendre wanted it as n-1 or why everybody else didn't have a problem with him shifting it, other than the fact that all the fundamentally important aspects of the function are still there so it's in a sense not really that important.

[Anonymous No.8469809]

>>8469784
the thing that baffles me is that the gamma function shows up in statistics and other field with a normal argument rather than (x+1) that would make the integers concide with n!

[Anonymous No.8469899]

>>8469665
The polygamma function is the logarithmic derivative of the gamma function and is closely related to the natural log. In a sense the polygamma function is like a discrete version of the natural log. The gamma function might be defined with the offset so that they coincide.
[math]\displaystyle \psi(x) - \psi(a) = \sum_{k=a}^{x-1} \frac{1}{k}[/math]
[math]\displaystyle \ln(x) - \ln(a) = \int_a^x \frac{1}{t} dt[/math]