How Do You Prove the Bessel Identity J-3/2(x)?

[yungman]

I have been working on this for a few days and cannot prove this:

J-3/2 (x)=$$\sqrt{\frac{2}{\pi x}}$$[$$\frac{-cos(x)}{x}$$ - sin(x) ]

Main reason is $$\Gamma$$(n-3/2+1) give a negative value for n=0 and possitive value for n=1,2,3... I cannot find a series representation of this gamma function.


Please advice me how to solve this problem. This is not a school homework.

thanks a million

Alan

 

[g_edgar]

Maybe instead of the series representation for the Bessel function: show that your right-hand side satisfies Bessel's differential equation, and has the proper initial values, so that it therefore equals the Bessel function required.

 

[matematikawan]

Main reason is $$\Gamma$$(n-3/2+1) give a negative value for n=0 and possitive value for n=1,2,3... I cannot find a series representation of this gamma function.


Please advice me how to solve this problem. This is not a school homework.

thanks a million

Alan

For n > 0 we apply this formula
$$\Gamma (n+1) = n \Gamma (n)$$

but if n < 0 we apply this formula
$$\Gamma (n)=\frac{\Gamma (n+1)}{n}$$

 

[yungman]

For n > 0 we apply this formula
$$\Gamma (n+1) = n \Gamma (n)$$

but if n < 0 we apply this formula
$$\Gamma (n)=\frac{\Gamma (n+1)}{n}$$

I know this formula! This is embarassing! How can I over looked this and spent 3 days on this...Even joined two more math forums! I even plug in the numbers and hope this is not that simple! I use

$$\Gamma (-3/2)=\frac{\Gamma (-1/2)}{-3/2}$$ all the time! Just never try with n in it!

Thanks a million...Even though you make me look really really bad!

Cheers.
Alan