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. Below, Γ(z) is the gamma function and ψ(z) is the digamma function.