INTERNATIONAL ACADEMY AMMAN
Extending the Domain of the Gamma Function
Math Exploration
Laila Hanandeh
11/10/2014
Table of Contents:
Aim 2
Factorials 2
The Zero Factorial 2
Deducing the Gamma Function 3
Working Out Example 6
Analytical Continuation 9
Gamma Function Graphs 10
Real Life Applications 11
Aim:
The Gamma Function is defined as an extension of the factorial function in which its argument is for complex and real numbers. (1) However, through my exploration I will determine a method to extend the domain of the gamma function to include complex numbers; this will be done through exploring the gamma function and the utilization of a process called analytical continuation.
However before proceeding,
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Г(8)=∫_0^∞▒〖t^(8-1) e^(-t) dt〗
Г(8)=∫_0^∞▒〖t^7 e^(-t) dt〗
∫_0^∞▒〖t^7 e^(-t) dt〗=├ -t^7 e^(-t) ┤| ∞¦0-∫_0^∞▒(7t^6 )(-e^(-t) )dt
∫_0^∞▒〖t^7 e^(-t) dt〗=├ -t^7 e^(-t) ┤| ∞¦0+7∫_0^∞▒〖t^6 e^(-t) dt〗
∫_0^∞▒〖t^7 e^(-t) dt〗=├ -t^7 e^(-t) ┤| ∞¦0-7├ t^6 e^(-t) ┤| ∞¦0-7∫_0^∞▒(6t^5 )(-e^(-t) )dt
∫_0^∞▒〖t^7 e^(-t) dt〗=├ -t^7 e^(-t) ┤| ∞¦0-7├ t^6 e^(-t) ┤| ∞¦0+42∫_0^∞▒〖t^5 e^(-t) dt〗
∫_0^∞▒〖t^7 e^(-t) dt〗=├ -t^7 e^(-t) ┤| ∞¦0-7├ t^6 e^(-t) ┤| ∞¦0 ├ -42t^5 e^(-t) ┤| ∞¦0-42∫_0^∞▒(〖5t〗^4 )(-e^(-t) )dt
∫_0^∞▒〖t^7 e^(-t) dt〗=├ -t^7 e^(-t) ┤| ∞¦0-7├ t^6 e^(-t) ┤| ∞¦0 ├ -42t^5 e^(-t) ┤| ∞¦0+210∫_0^∞▒〖t^4 e^(-t) dt〗
∫_0^∞▒〖t^7 e^(-t) dt〗=├ -t^7 e^(-t) ┤| ∞¦0-7├ t^6 e^(-t) ┤| ∞¦0 ├ -42t^5 e^(-t) ┤| ∞¦0 ├ -210t^4 e^(-t) ┤| ∞¦0-210∫_0^∞▒(〖4t〗^3 )(-e^(-t) )dt
∫_0^∞▒〖t^7 e^(-t) dt〗=├ -t^7 e^(-t) ┤| ∞¦0-7├ t^6 e^(-t) ┤| ∞¦0 ├ -42t^5 e^(-t) ┤| ∞¦0 ├ -210t^4 e^(-t) ┤| ∞¦0+840∫_0^∞▒〖t^3 e^(-t) dt〗
∫_0^∞▒〖t^7 e^(-t) dt〗=├ -t^7 e^(-t) ┤| ∞¦0-7├ t^6 e^(-t) ┤| ∞¦0 ├ -42t^5 e^(-t) ┤| ∞¦0 ├ -210t^4 e^(-t) ┤| ∞¦0 ├ -840t^3 e^(-t) ┤| ∞¦0-840∫_0^∞▒(〖3t〗^2 )(-e^(-t)
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Through such a process, the domain of the gamma function would extend to include a complex domain where x>0. In which all negative numbers are once again not part of the domain.
This is explained below,
Since
Г(n)=Г(n+1)/n
And
Г(n)=(n-1)!
Therefore,
Г(n+1)=n(n-1)!
The domain of the function Г(n+1) is {n:n∈├ C} {x:┤x>├ -1}┤ in which C denotes the complex plane while x denotes the real number plane. Thus, one can see that the domain of the function Г(n) can be extended to include{n:n∈├ C}┤, with x>-1 and x≠0.
This function could also be represented as:
Г(n)=Г(n+2)/(n(n+1))
The above function has a domain of{n:n∈├ C}┤, with x>-2 and x≠0 and x≠-1. Therefore, once again the domain of the gamma function is extended.
One can see that through this a certain pattern arises which can be presented more clearly through the following:
Г(n)=Г(n+m)/((n+m-1)(n+m-2)…(n+1)n), {m:m∈├ Ƶ^+ }┤
This results in the domain of the function to extended to include complex number in which {n:n∈C,n≠0,├ Ƶ^- }┤, this is otherwise known as R(n).
Graphs of the Gamma Function:
The following graph shows the real part of an analytically continued gamma