In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric. More generally, introduced Whittaker functions of reductive groups over local fields, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL2(R).
Whittaker's equation is
| d2w | +\left(- | |
| dz2 |
| 1 | + | |
| 4 |
| \kappa | + | |
| z |
| 1/4-\mu2 | |
| z2 |
\right)w=0.
M\kappa,\mu\left(z\right)=\exp\left(-z/2\right)z\mu+\tfrac{1{2}}M\left(\mu-\kappa+\tfrac{1}{2},1+2\mu,z\right)
W\kappa,\mu\left(z\right)=\exp\left(-z/2\right)z\mu+\tfrac{1{2}}U\left(\mu-\kappa+\tfrac{1}{2},1+2\mu,z\right).
The Whittaker function
W\kappa,\mu(z)
Whittaker functions appear as coefficients of certain representations of the group SL2(R), called Whittaker models.
