Problem in Gama function;can you answer soon.

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Hello i am very happy to sent my firest question,so i am very happy from forum.
my problem is : prove that
Printed equation: Gamma(-m + 1/2) = (-1)^m 2^m sqrt(pi) / (1.3.5.... (2m - 1))

using
Functional equation of the gamma function: Γ(x + 1) = xΓ(x).



I try by:
let
Typeset equation: Gamma(x) = Gamma(x+1) divided by x, where Gamma is the gamma function.
 
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  • Typeset formula: Gamma(-m + 1/2) = ((-1)^m 2^m sqrt(pi)) / (1.3.5.... (2m - 1)), a Gamma function identity for negative half-integer arguments.
    right&space;)^{m}2^{m}\sqrt{\pi&space;}}{1.3.5....\left&space;(&space;2m-1&space;\right&space;)}.gif
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quantum220 said:
Hello i am very happy to sent my firest question,so i am very happy from forum.
my problem is :
prove that
Formula for Gamma(-m + 1/2) equal to (-1)^m 2^m sqrt(pi) divided by 1.3.5.... (2m - 1).

The pi term comes from the fact that Gamma(1/2) = SQRT(pi). The rest you can deduce from using the relationship Gamma(x + 1) = x * Gamma(x). Basically because they are negative you are going to get switching signs since every step is multiplying by a negative number and the rest can be obtained from the definition. (I'm assuming m is an integer of course)
 
If m= 0, that says that [itex]\Gamma(1/2)= \sqrt{\pi}[/itex] which is true.

Now, use [itex]\Gamma(x+ 1)= x\Gamma(x)[itex]to prove the rest by induction on m.[/itex][/itex]