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?
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]}}*)
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.
{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]}*)
I truly appreciate your help in resolving the problem.