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Solutions to the the Kelvin differential equation in R:

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kelvin

Solutions to the the Kelvin differential equation in R

by Andrew J Barbour

Calculation of solutions to the Kelvin Differential Equation using Bessel functions namely `BesselJ` and `BesselK` from the Bessel package.

Background information

The following is taken from Wolfram:

Kelvin defined the Kelvin functions bei and ber according to

``````ber_v(x) + i*bei_v(x)
=	J_v(x*exp(2*pi*i/4))
=	exp(v*pi*i)*J_v(x*exp(-pi*i/4))
=	exp(v*pi*i/2)*I_v(x*exp(pi*i/4))
=	exp(3*v*pi*i/2)*I_v(x*exp(-3*pi*i/4))
``````

where J_v(x) is a Bessel function of the first kind and I_v(x) is a modified Bessel function of the first kind. These functions satisfy the Kelvin differential equation.

Similarly, the functions kei and ker by

``````ker_v(x) + i*kei_v(x) = exp(-v*pi*i/2)*K_v(x*exp(pi*i/4))
``````

where K_v(x) is a modified Bessel function of the second kind. For the special case v=0,

``````J_0(i*sqrt(i)*x)
=	J_0(sqrt(2)*(i-1)*x/2)
=	ber(x) + i*bei(x)
``````

Solutions to the the Kelvin differential equation in R:

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