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Mathematics Calculus Wolfram Language

How to turn BesselJ and BesselY into BesselI and BesselK?

Jacques Ou

Posted 3 years ago

Mathematica gives the solution of (4 R^2 + Rv^2) Cs1[R, T] - 4 R ( \!\(\SuperscriptBox[\(Cs1\), TagBox[ RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[R, T] + R \!\(\SuperscriptBox[\(Cs1\), TagBox[ RowBox[{"(", RowBox[{"2", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[R, T])== 0 with BesselJ[Rv/2, -I R] C[1] + BesselY[Rv/2, -I R] C[2] How to turn the solution as above into BesselI and BesselK since the equation is the modified Bessel equation.

Mathematica gives the solution of

(4 R^2 + Rv^2) Cs1[R, T] - 4 R (
\!\(\*SuperscriptBox[\(Cs1\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R, T] + R 
\!\(\*SuperscriptBox[\(Cs1\), 
TagBox[
RowBox[{"(", 
RowBox[{"2", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[R, T])== 0

with

BesselJ[Rv/2, -I R] C[1] + BesselY[Rv/2, -I R] C[2]

How to turn the solution as above into BesselI and BesselK since the equation is the modified Bessel equation.

POSTED BY: Jacques Ou

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Jacques Ou

Posted 3 years ago

I am curious about why Mathematica give the modified Bessel equation the solution of Bessel functions rather than modified Bessel functions? In[53]:= eq15 = DSolve[y''[x] + 1/x y'[x] - (1 + v^2/x^2) y[x] == 0, y[x], x] Out[53]= {{y[x] -> BesselJ[v, -I x] C[1] + BesselY[v, -I x] C[2]}} BesselI[Rv/2, R] C[1] + BesselK[Rv/2, R] C[2] is the solution indeed of modified Bessel equation.

POSTED BY: Jacques Ou

Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 3 years ago

POSTED BY: Mariusz Iwaniuk

Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 3 years ago

BesselJ[Rv/2, -I R] C[1] + BesselY[Rv/2, -I R] C[2] // FunctionExpand (-((2 (-I R)^(-Rv/2) R^(Rv/2) BesselK[Rv/2, R] C[ 2])/\[Pi]) + (-I R)^(-Rv/2) R^(-Rv/2) BesselI[Rv/2, R] ((-I R)^Rv C[1] + (-I R)^Rv C[2] Cot[(\[Pi] Rv)/2] - R^Rv C[2] Csc[(\[Pi] Rv)/2]))

BesselJ[Rv/2, -I R] C[1] + BesselY[Rv/2, -I R] C[2] // FunctionExpand

(*-((2 (-I R)^(-Rv/2) R^(Rv/2) BesselK[Rv/2, R] C[
   2])/\[Pi]) + (-I R)^(-Rv/2) R^(-Rv/2)
   BesselI[Rv/2, 
   R] ((-I R)^Rv C[1] + (-I R)^Rv C[2] Cot[(\[Pi] Rv)/2] - 
    R^Rv C[2] Csc[(\[Pi] Rv)/2])*)

POSTED BY: Mariusz Iwaniuk

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