کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4635858 | 1340715 | 2007 | 23 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions](/preview/png/4635858.png)
In 1971, T.J. Osler propose a generalization of Taylor’s series of f(z ) in which the general term is [Dz0-ban+γf(z0)](z-z0)an+γ/Γ(an+γ+1), where 0 < a ⩽ 1, b ≠ z0 and γ is an arbitrary complex number and Dzα is the fractional derivative of order α. In this paper, we present a new expansion of an analytic function f(z ) in RR in terms of a power series θ(t) = tq(t), where q(t) is any regular function and t is equal to the quadratic function [(z − z1)(z − z2)] , z1 ≠ z2, where z1 and z2 are two points in RR and the region of validity of this formula is also deduced.To illustrate the concept, if q(t) = 1, the coefficient of (z − z1)n(z − z2)n in the power series of the function (z − z1)α(z − z2)βf(z ) is Dz1-z2-α+n[f(z1)(z1-z2)β-n-1(z1-z2+z-w)]|w=z1/Γ(1-α+n) where α and β are arbitrary complex numbers. Many special forms are examined and some new identities involving special functions and integrals are obtained.
Journal: Applied Mathematics and Computation - Volume 187, Issue 1, 1 April 2007, Pages 507–529