Baer function

Baer functions ${\displaystyle B_{p}^{q}(z)}$ and ${\displaystyle C_{p}^{q}(z)}$, named after Karl Baer,[1] are solutions of the Baer differential equation

${\displaystyle {\frac {d^{2}B}{dz^{2}}}+{\frac {1}{2}}\left[{\frac {1}{z-b}}+{\frac {1}{z-c}}\right]{\frac {dB}{dz}}-\left[{\frac {p(p+1)z+q(b+c)}{(z-b)(z-c)}}\right]B=0}$

which arises when separation of variables is applied to the Laplace equation in paraboloidal coordinates. The Baer functions are defined as the series solutions about ${\displaystyle z=0}$ which satisfy ${\displaystyle B_{p}^{q}(0)=0}$, ${\displaystyle C_{p}^{q}(0)=1}$.[2] By substituting a power series Ansatz into the differential equation, formal series can be constructed for the Baer functions.[3] For special values of ${\displaystyle p}$ and ${\displaystyle q}$, simpler solutions may exist. For instance,

${\displaystyle B_{0}^{0}(z)=\ln \left[{\frac {z+{\sqrt {(z-b)(z-c)}}-(b+c)/2}{{\sqrt {bc}}-(b+c)/2}}\right]}$

Moreover, Mathieu functions are special-case solutions of the Baer equation, since the latter reduces to the Mathieu differential equation when ${\displaystyle b=0}$ and ${\displaystyle c=1}$, and making the change of variable ${\displaystyle z=\cos ^{2}t}$.

Like the Mathieu differential equation, the Baer equation has two regular singular points (at ${\displaystyle z=b}$ and ${\displaystyle z=c}$), and one irregular singular point at infinity. Thus, in contrast with many other special functions of mathematical physics, Baer functions cannot in general be expressed in terms of hypergeometric functions.

The Baer wave equation is a generalization which results from separating variables in the Helmholtz equation in paraboloidal coordinates:

${\displaystyle {\frac {d^{2}B}{dz^{2}}}+{\frac {1}{2}}\left[{\frac {1}{z-b}}+{\frac {1}{z-c}}\right]{\frac {dB}{dz}}+\left[{\frac {k^{2}z^{2}-p(p+1)z-q(b+c)}{(z-b)(z-c)}}\right]B=0}$

which reduces to the original Baer equation when ${\displaystyle k=0}$.