Mellin transform of the Bessel function
We show the following result ∫∞0(x2)sJs(x)dx=Γ(s)
Proof
We will use the Ramanujan's master theorem to prove this result: Assume that the function f has an expansion of the form f(x)=∞∑k=0(ϕ(k)k!)(−x)k
for some analytic function ϕ(k), then the Mellin transform of f(x) is given by
{Mf(x)}=∫∞0xs−1f(x)dx=Γ(s)ϕ(−s)
Back to our problem, recall the expansion series of the Bessel function:
Js(x)=(x2)s∞∑j=0(−x2/4)jj!Γ(1+j+s)
Then
∫∞0(x2)s−1Js(x)dx=∫∞0(x2)2s−1∞∑j=0(−x2/4)jj!Γ(1+j+s)dx
let w=x24 then dw=x2dx, therefore
∫∞0(x2)2s−1∞∑j=0(−x2/4)jj!Γ(1+j+s)dx=∫∞0ws−1∞∑j=0(−w)jj!Γ(1+j+s)dw
Clearly f(w)=∞∑j=0(−w)jj!Γ(1+j+s) and ϕ(j)=1Γ(1+j+s)
By Ramanujan's master theorem
:
∫∞0ws−1∞∑j=0(−w)jj!Γ(1+j+s)dw=Γ(s)ϕ(−s)=Γ(s)Γ(1)=Γ(s)
Then
∫∞0(x2)sJs(x)dx=Γ(s)
No comments:
Post a Comment