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Consider the following function

h[j_] := Sqrt[1 + l^2*j^2]/j

Then (the integral exists in the Hadamard sense)

FourierSinTransform[h[j], j, x]
-(MeijerG[{{1}, {}}, {{-(1/2), 1/2}, {0}}, x^2/(4*l^2)]/(2*Sqrt[2*Pi]))


FullSimplify[FourierSinTransform[%, x, j], {l > 0, j > 0}]
Sqrt[l^2 + 1/j^2]

Can I trust the first result?

Thanks in adance for any response.

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