- #1
LagrangeEuler
- 717
- 20
I have question regarding gamma function. It is concerning ##\Gamma## function of negative integer arguments.
Is it ##\Gamma(-1)=\infty## or ##\displaystyle \lim_{x \to -1}\Gamma(x)=\infty##? So is it ##\Gamma(-1)## defined or it is ##\infty##? This question is mainly because of definition of Bessel function
[tex] J_p(x)=\sum^{\infty}_{n=0}\frac{(-1)^n}{\Gamma(n+1)\Gamma(n+p+1)}(\frac{x}{2})^{2n+p}[/tex]
For example there is relation
[tex]J_{-p}(x)=(-1)^pJ_p(x) [/tex]
What is happening with ##J_{-2}##
[tex] J_{-2}(x)=\sum^{\infty}_{n=0}\frac{(-1)^n}{\Gamma(n+1)\Gamma(n-1)}(\frac{x}{2})^{2n-2}[/tex]
for term ##n=0##?
Is it ##\Gamma(-1)=\infty## or ##\displaystyle \lim_{x \to -1}\Gamma(x)=\infty##? So is it ##\Gamma(-1)## defined or it is ##\infty##? This question is mainly because of definition of Bessel function
[tex] J_p(x)=\sum^{\infty}_{n=0}\frac{(-1)^n}{\Gamma(n+1)\Gamma(n+p+1)}(\frac{x}{2})^{2n+p}[/tex]
For example there is relation
[tex]J_{-p}(x)=(-1)^pJ_p(x) [/tex]
What is happening with ##J_{-2}##
[tex] J_{-2}(x)=\sum^{\infty}_{n=0}\frac{(-1)^n}{\Gamma(n+1)\Gamma(n-1)}(\frac{x}{2})^{2n-2}[/tex]
for term ##n=0##?