In applied mathematics, the Kelvin functions berν(x) and beiν(x) are the real and imaginary parts, respectively, of where x is real, and Jν(z), is the νth order Bessel function of the first kind.
Similarly, the functions kerν(x) and keiν(x) are the real and imaginary parts, respectively, of where Kν(z) is the νth order modified Bessel function of the second kind.
These functions are named after William Thomson, 1st Baron Kelvin.
With the exception of bern(x) and bein(x) for integral n, the Kelvin functions have a branch point at x = 0.
For integers n, bern(x) has the series expansion where Γ(z) is the gamma function.
Plot of the Kelvin function ber(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
ber(
x
) for
x
between 0 and 20.
for
x
between 0 and 50.
Plot of the Kelvin function bei(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
bei(
x
) for
x
between 0 and 20.
for
x
between 0 and 50.
Plot of the Kelvin function ker(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
ker(
x
) for
x
between 0 and 14.
for
x
between 0 and 50.
Plot of the Kelvin function kei(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
