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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

3 Replies

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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

EQ = y''[x] + 1/x y'[x] - (1 + v^2/x^2) y[x] == 0; SOL = DSolve[EQ, y, x] {{y -> Function[{x}, Evaluate@(SOL[[1, 1, 2, 2]] // FunctionExpand)]}} EQ /. SOL // FullSimplify(We can check is solution is True) (True) Yes modified Bessel function are solution for this equation. "Why Mathematica give the modified Bessel equation the solution of Bessel functions rather than modified Bessel functions?", I do not know the answer to this question.

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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