Processing math: 100%

Thursday, June 24, 2021

Application of Ramanujan's master theorem

Bessel function

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)}=0xs1f(x)dx=Γ(s)ϕ(s)
Back to our problem, recall the expansion series of the Bessel function: Js(x)=(x2)sj=0(x2/4)jj!Γ(1+j+s)
Then 0(x2)s1Js(x)dx=0(x2)2s1j=0(x2/4)jj!Γ(1+j+s)dx
let w=x24 then dw=x2dx, therefore 0(x2)2s1j=0(x2/4)jj!Γ(1+j+s)dx=0ws1j=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 : 0ws1j=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

Series of the day

Series involving the digamma and the zeta functions The sum 1(n+1)pnq ...