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:
“Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
—Ralph Waldo Emerson (1803–1882)
“Hummingbird
stay for a fractional sharp
sweetness, and’s gone, can’t take
more than that.”
—Denise Levertov (b. 1923)
“The universal principle of etymology in all languages: words are carried over from bodies and from the properties of bodies to express the things of the mind and spirit. The order of ideas must follow the order of things.”
—Giambattista Vico (1688–1744)