Generalization Via Hypergeometric Functions
These functions may actually be defined for general complex parameters and argument:
where is the gamma function and is the hypergeometric function
They are called the Legendre functions when defined in this more general way. They satisfy the same differential equation as before:
Since this is a second order differential equation, it has a second solution, defined as:
and both obey the various recurrence formulas given previously.
Read more about this topic: Associated Legendre Polynomials
Famous quotes containing the word functions:
“Nobody is so constituted as to be able to live everywhere and anywhere; and he who has great duties to perform, which lay claim to all his strength, has, in this respect, a very limited choice. The influence of climate upon the bodily functions ... extends so far, that a blunder in the choice of locality and climate is able not only to alienate a man from his actual duty, but also to withhold it from him altogether, so that he never even comes face to face with it.”
—Friedrich Nietzsche (1844–1900)