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Modified Bessel differential equation?

Xin Ning

Posted 2 years ago

I have a very basic and simple question that I just cannot figure out myself. I spent several hours trying and looking in the help, but no dice. How can BesselJ and BesselY be transformed into BesselI and BesselK in Mathematica. Examples are as follows: the result I calculated through Mathematica is the form of BesselJ and BesselY, but I need is the form of BesselI and BesselK. But how do I do that?

POSTED BY: Xin Ning

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Xin Ning

Posted 2 years ago

Thank you very much! Can the final form be transformed into a form without imaginary number i? I use Euler's formula and find that there is still an imaginary number i in the final form.

POSTED BY: Xin Ning

Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 2 years ago

{Z[R, T] -> Simplify[ComplexExpand[(Z[R, T] /. eq2[[1, 1]]) // Re], Assumptions -> {\[Alpha] > 0, R > 0, T > 0}] // Expand} (* {Z[R, T] -> BesselI[\[Alpha], R] Cos[(\[Pi] \[Alpha])/2] C[1][T] - ( 2 BesselK[\[Alpha], R] Cos[(\[Pi] \[Alpha])/2] C[2][T])/\[Pi] - BesselI[\[Alpha], R] Sin[(\[Pi] \[Alpha])/2] C[2][T]}*)

{Z[R, T] -> 
   Simplify[ComplexExpand[(Z[R, T] /. eq2[[1, 1]]) // Re], 
    Assumptions -> {\[Alpha] > 0, R > 0, T > 0}] // Expand}


 (* {Z[R, T] -> 
   BesselI[\[Alpha], R] Cos[(\[Pi] \[Alpha])/2] C[1][T] - (
    2 BesselK[\[Alpha], R] Cos[(\[Pi] \[Alpha])/2] C[2][T])/\[Pi] - 
    BesselI[\[Alpha], R] Sin[(\[Pi] \[Alpha])/2] C[2][T]}*)

POSTED BY: Mariusz Iwaniuk

Xin Ning

Posted 2 years ago

I truly appreciate your help in resolving the problem.

POSTED BY: Xin Ning

Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 2 years ago

Try: eq2 = FullSimplify[DSolve[eq1, Z[R, T], {R, T}] // FunctionExpand, Assumptions -> {\[Alpha] > 0, R > 0, T > 0}] // Expand ({{Z[R, T] -> E^(-(1/2) I \[Pi] \[Alpha]) BesselI[\[Alpha], R] C[1][T] - I E^(-(1/2) I \[Pi] \[Alpha]) BesselI[\[Alpha], R] C[2][T] - ( 2 E^((I \[Pi] \[Alpha])/2) BesselK[\[Alpha], R] C[2][T])/\[Pi]}})

Try:

eq2 = FullSimplify[DSolve[eq1, Z[R, T], {R, T}] // FunctionExpand, 
 Assumptions -> {\[Alpha] > 0, R > 0, T > 0}] // Expand

(*{{Z[R, T] -> 
E^(-(1/2) I \[Pi] \[Alpha]) BesselI[\[Alpha], R] C[1][T] - 
I E^(-(1/2) I \[Pi] \[Alpha]) BesselI[\[Alpha], R] C[2][T] - (
2 E^((I \[Pi] \[Alpha])/2) BesselK[\[Alpha], R] C[2][T])/\[Pi]}}*)

POSTED BY: Mariusz Iwaniuk

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