# How to find the roots of a transcendental equation that depends on Bessel functions? [duplicate]

I am very new in wolfram mathematica, so i am having trouble in trying to find the roots for this transcendental equation. I am looking for roots of m. Acctually the first non zero value for m. Here J is the Bessel function on first kind and N is the Bessel function of second kind. I have tried some codes but it did not work.

• Please type your equation in Mathematica format. Do not post a picture. – yarchik Jul 18 '20 at 18:58
• Try Solve[eqn && 11000 < m < 100000, m]. – Michael E2 Jul 18 '20 at 21:52

In Mathematica, a Bessel function of the second kind is BesselY

Clear["Global*"]

l = 10^-4; L = 10^-6;

eqn = -BesselY[2, m*L]*BesselJ[1, m*L]*
BesselJ[2, m*(l + L)]*BesselY[1, m*(l + L)] -
BesselJ[2, m*L]*BesselJ[1, m*L]*
BesselY[2, m*(l + L)]*BesselJ[1, m*(l + L)] +
BesselY[2, m*L]*BesselY[1, m*L]*
BesselJ[2, m*(l + L)]*BesselJ[1, m*(l + L)] +
BesselJ[2, m*L]*BesselJ[1, m*L]*
BesselY[2, m*(l + L)]*BesselY[1, m*(l + L)] == 0;

LogLinearPlot[Evaluate@eqn[[1]], {m, 0, 100000},
PlotPoints -> 100, MaxRecursion -> 2]


LogLinearPlot[Evaluate@eqn[[1]], {m, 3*^4, 100000}]


The plot provides initial estimates to use with FindRoot

sol = FindRoot[eqn, {m, #}] & /@ {3.5*^4, 5*^4, 7*^4, 10^5}

{{m -> 37949.1}, {m -> 50847.7}, {m -> 69498.8}, {m -> 100805.}}
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