Legendre Functions of Fractional Order
Legendre functions of fractional order exist and follow from insertion of fractional derivatives as defined by fractional calculus and non-integer factorials (defined by the gamma function) into the Rodrigues' formula. The resulting functions continue to satisfy the Legendre differential equation throughout (−1,1), but are no longer regular at the endpoints. The fractional order Legendre function Pn agrees with the associated Legendre polynomial P0
n.
Read more about this topic: Legendre Polynomials
Famous quotes containing the words functions, fractional and/or order:
“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)
“Hummingbird
stay for a fractional sharp
sweetness, and’s gone, can’t take
more than that.”
—Denise Levertov (b. 1923)
“It is the international system of currency which determines the totality of life on this planet. That is the natural order of things today. That is the atomic, and sub-atomic, and galactic
structure of things today. And you have meddled with the primal forces of nature! And you will atone! Am I getting through to you, Mr. Beale?”
—Paddy Chayefsky (1923–1981)