modified Bessel function

The modified Bessel function is the solution of the differential equation:

x² y '' + x y' - (x² + n²) y = 0

which is slightly different from the Bessel differential equation that defines the Bessel function.

There are two solutions:

• modified Bessel function of the first kind I_n(x)
  The function can be written in terms of the Bessel function of the first kind J_n(x) as follows:
  I_n(x) = i^-n J_n(ix).
  
• modified Bessel function of the second kind K_n(x)
  The function can be written in terms of the Bessel function of the first kind J_n(x) and the Bessel function of the second kind N_n(x) as follows: K_n(x) = ½ πiⁿ⁺¹ (J_n(ix) + i N_n(ix))
  
  The function bears also the following names:
  • hyperbolic Bessel function
  • Basset function
  • modified Bessel function of the third kind
  • Macdonald function

  
  
  From the modified Bessel function two modified Kelvin functions ker_n(x) and kei_n(x) can be derived, in the following way:

  ker_n(x) + i kei_n(x) = e^-nπi/2 K_n(x e ^πi/4)