Definition:Kelvin Function

Definition

The Kelvin functions are defined as the real parts and imaginary parts of the Bessel functions of the first and modified second kind.

Ber Function

Let $J_n$ denote the Bessel function of the first kind.

The Ber function is defined as:

$\map {\Ber_n} x = \map \Re {\map {J_n} {x \map \exp {\dfrac {3 \pi i} 4} } }$

where:

$\exp$ denotes the exponential function

$x$ is real

$\map \Re z$ denotes the real part of $z$.

Bei Function

Let $J_n$ denote the Bessel function of the first kind.

The Bei function is defined as:

$\map {\Bei_n} x = \map \Im {\map {J_n} {x \map \exp {\dfrac {3 \pi i} 4} } }$

where:

$\exp$ denotes the exponential function

$x$ is real

$\map \Im z$ denotes the imaginary part of $z$.

Ker Function

Let $K_n$ denote the modified Bessel function of the second kind.

The Ker function is defined as:

$\map {\Ker_n} x = \map \Re {\map {K_n} {x \map \exp {\dfrac {\pi i} 4} } }$

where:

$\exp$ denotes the exponential function

$x$ is real

$\map \Re z$ denotes the real part of $z$.

Kei Function

Let $K_n$ denote the modified Bessel function of the second kind.

The Kei function is defined as:

$\map {\Kei_n} x = \map \Im {\map {K_n} {x \map \exp {\dfrac {\pi i} 4} } }$

where:

$\exp$ denotes the exponential function

$x$ is real

$\map \Im z$ denotes the imaginary part of $z$.

Also see

• Results about the Kelvin functions can be found here.

Source of Name

This entry was named for Lord Kelvin.