Help with Bessel Function

Questions
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1

I need help solving for a variable in modifed bessel function of the second kind and I have no idea. Can anyone help? Below is the equation and and what I’ve put in Mathematica.

spielmanandgorenequation.png
spielmanandgorenequation.png211×149 1.01 KB

I need to solve for K.
d = 0.000012401
a = 0.0382

solve[0.5 + ((2 [Sqrt]x)/0.000012401) (BesselK[1, (0.000012401/(2 [Sqrt]x))]/BesselK[0, (0.000012401/(2 [Sqrt]x))]) == 1/(4*0.0382), x]

2

Hello D. Moore,

here is your equation to start from:
eq10 = 1/2 + ((2 Sqrt[k])/d) BesselK[1, (d/(2 Sqrt[k]))] /
BesselK[0, (d/(2 Sqrt[k]))] == 1/(4 a);

It can be rewritten by substituting
(d/(2 Sqrt[k])) → z :
eq11 = 1/2 + (1/z) BesselK[1, z] /BesselK[0, z] == 1/(4 a);

You want to solve it numerically using FindRoot, therefore you have to reformat the equation in a way that the right handside is equal to zero. And you replace the symbol a by its value:
eq12 = eq11[[1]] - 1/(4 a) /. { a → 0.0382}

Now solve using FindRoot:
sol1 = FindRoot[eq12, {z, 1.}]

To get back to k, you enter sol1 in this equation:
sol2 = k → (d / (2 z))^2 /. sol1 /. d → 0.000012401

Finally you want to check that the result is correct:
eq10 /. sol2 /. { a → 0.0382, d → 0.000012401}

Regards,

Michael

3

k_n is a modified Bessel function of the third kind, proportional to K_{n+1/2). Also Sqrt is coded incorrectly. Here’s what I get for numerical solution of your problem:

In[14]:= d = 0.000012401; a = 0.0382;
FindRoot[0.5 + ((2 Sqrt[x])/
      0.000012401) (BesselK[3/2, (0.000012401/(2 Sqrt[x]))]/
      BesselK[1/2, (0.000012401/(2 Sqrt[x]))]) == 
  1/(4*0.0382), {x, .1}]

Out[15]= {x -> 1.55154*10^-10 - 9.43222*10^-21 I}

In[16]:= Chop[%]

Out[16]= {x -> 1.55154*10^-10}
4

Thanks so much