The modified Bessel function is the solution of the differential equation:
x^{2} y '' + x y' - (x^{2} + n^{2}) y = 0
which is slightly different from the Bessel differential equation that defines the Bessel function.
There are two solutions:
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})