Kelvin functions
In applied mathematics, the Kelvin functions berν(x) and beiν(x) are the real and imaginary parts, respectively, of J ν ( x e 3 π i 4 ) , {\displaystyle J_{\nu }\left(xe^{\frac {3\pi i}{4}}\right),\,} 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 e − ν π i / 2 K ν ( x e π i 4 ) , {\displaystyle e^{-\nu \pi i/2}K_{\nu }\left(xe^{\frac {\pi i}{4}}\right),\,} where Kν(z) is the νth order modified Bessel function of the second kind.