The Gamma function,
G(n)= Integral from 0 to infinity [Exp(e^-x^n)]dx
is of the great mathematical and physical importance.
It can be calculated without numerical integration (for practical purposes) via its mathematical and physical properties:
i-minimum of Gamma occurs at x = 1.4616321 and the corresponding value of Gamma(x) is 0.8856032.
ii-Gamma(1.)=Gamma(2.)=1.
iii-Gamma(x)=(x-1.) !
A simple preliminary approach that gives the value of Gamma(x) with an error less than 0.001 is the second-order polynomial expression for the factorial x,
(1.-0.46163*x+0.46163*x*x),x element of [0,1].
For example, this gives:
G(10.5)=11877478.
vs the value of 11899423.084) given by numerical integration.
and Gamma(1.4616)= 0.88527 vs 0.8856032.
e.g., "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables", 1964, by Abramowitz and Stegun, might help with a chapter for Gamma function properties and formulas
Chapter 6 of Abramowitz & Stegun - available free online just Google it...
The Gamma function G(n) is well defined for any positive value of n,
G(n)= Integral from 0 to infinity [Exp(e^-x^n)]dx . . . . . (1)
Needless to say, it is of great mathematical and physical importance.
However, for practical purposes, it can be calculated using an adequate closed-form polynomial without going through complicated numerical integration.
The required closed form solution must retain its mathematical and physical properties, namely:
i-minimum of Gamma occurs at x = 1.4616321 and the corresponding value of Gamma(x) is 0.8856032.
ii-Gamma(1.)=Gamma(2.)=1.
iii-Gamma(x)=(x-1.) !
iv-The recurrence relation ,Gamma (x)=x. Gamma (x-1)
We propose a simple preliminary approach that gives the required value of Gamma(x) with an error less than 0.001. The proposed preliminary approach must satisfy conditions i-iv,
but since the factorial function x! is not yet defined for negative numbers (x
Yes, Gamma function can be calculated without numerical integration,
through an infinite product expression - seen here
https://math.stackexchange.com/questions/2880150/proving-that-the-gamma-function-infinite-product-definition-extends-its-integral
or, here
https://en.wikipedia.org/wiki/Gamma_function#Euler's_definition_as_an_infinite_product
... the infinite product expression is valid for all complex numbers z except for the negative integers
https://www.google.com/search?q=gamma+function+plot
Good luck
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