Problem in Gama function;can you answer soon.

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SUMMARY

The discussion centers on proving the relationship involving the Gamma function, specifically that Gamma(1/2) equals the square root of pi (SQRT(pi)). The user highlights the recursive property of the Gamma function, defined as Gamma(x + 1) = x * Gamma(x), and emphasizes the importance of induction for proving further relationships. The proof relies on the fact that for integer values of m, the signs alternate due to multiplication by negative numbers.

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  • Understanding of the Gamma function and its properties
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quantum220
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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
right&space;)^{m}2^{m}\sqrt{\pi&space;}}{1.3.5....\left&space;(&space;2m-1&space;\right&space;)}.gif

using
gif.gif



I try by:
let
gif.gif
 

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  • right&space;)^{m}2^{m}\sqrt{\pi&space;}}{1.3.5....\left&space;(&space;2m-1&space;\right&space;)}.gif
    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
right&space;)^{m}2^{m}\sqrt{\pi&space;}}{1.3.5....\left&space;(&space;2m-1&space;\right&space;)}.gif

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]
 

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