Laplace transform

wtmoore

1. The problem statement, all variables and given/known data
I'm trying to find the Laplace transform of tJ''₀(t), it's from bessels equation, but that doesn't matter too much at the moment, I just need to integrate (e^-st)*t*J''₀(t) but am unsure how to go about this with the J''₀(t) in there.

Ray Vickson

Maple 14 gets an answer:
t*diff(BesselJ(0,t),t$2): <---second derivative of J0(t), times t

f:=expand(%);
f := -t BesselJ(0, t) + BesselJ(1, t)

L:=int(f*exp(-s*t),t=0..infinity);

L:= [1 + 2s^2 + s^4 - (s^2 + 2s)*sqrt(1 + s^2)]/ (1+s^2)^2

Note: Maple used the DE for J0 and recursions to express J0'' in terms of J0 and J1, then it integrated that.

RGV

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wtmoore	Laplace transform Tweet 1. The problem statement, all variables and given/known data I'm trying to find the Laplace transform of tJ''₀(t), it's from bessels equation, but that doesn't matter too much at the moment, I just need to integrate (e^-st)tJ''₀(t) but am unsure how to go about this with the J''₀(t) in there.

Ray Vickson Recognitions: Homework Help	Maple 14 gets an answer: tdiff(BesselJ(0,t),t$2): <---second derivative of J0(t), times t f:=expand(%); f := -t BesselJ(0, t) + BesselJ(1, t) L:=int(fexp(-st),t=0..infinity); L:= [1 + 2s^2 + s^4 - (s^2 + 2s)sqrt(1 + s^2)]/ (1+s^2)^2 Note: Maple used the DE for J0 and recursions to express J0'' in terms of J0 and J1, then it integrated that. RGV