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I really appreciate if anyone could indicate me how to handle this inverse Laplace transformation (ILT):

L

where

a(s)=(s

c

I searched some literatures regarding the ILT of Exp funtions but no such form. I used some general formulas of ILT and basic expressions with exponential functions, and finally the result involves an integral of the product of several functions including BesselJ_0 function and exponential function, which seems hard to integrate. Im also wondering is there any rule to judge whether the ILT of a function can be obtained analytically or not? If yes for this function, how to do it? Many thanks for any help!

L

^{-1}[Exp(-c_{0}*Sqrt(a(s)))/Sqrt(a(s))]where

a(s)=(s

^{2}+c_{1}s)/(c_{2}s+c_{3})c

_{0},c_{1},c_{2},c_{3}are all constants.I searched some literatures regarding the ILT of Exp funtions but no such form. I used some general formulas of ILT and basic expressions with exponential functions, and finally the result involves an integral of the product of several functions including BesselJ_0 function and exponential function, which seems hard to integrate. Im also wondering is there any rule to judge whether the ILT of a function can be obtained analytically or not? If yes for this function, how to do it? Many thanks for any help!

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