Backward compatibility with factorial

Special Functions of the Show that Γ(1) = 1 Fractional Calculus

© Igor Podlubny, 1999-2007 1 4

. . . from integer to non-integer . . .

- -2 -1 0 1 2

xn = x x . . . x · ·n · n n ln x Euler’s Gammax function!= e "# $ Simple poles at

Factorial: n! = 1 2 3 . . . (n 1) n, · · · · − · n! = n ∞(n 1)! Start Γ(x) = · e −ttx 1dt, x > 0, 0! = 1 − − !! "" %0 ! " Γ(n + 1) = 1 2 3 . . . n = n! 7 / 90 Leonhard Euler (1729): · · · · Back Full screen TheClose second integral defines an entire function of z. End consider this 2 5

Plot of !(x) for -5

10

8

6

4

2

(x) 0 !

!2 bounded !4 !6

!8

!10 must be x = Re(z) > 0 !5 !4 !3 !2 !1 0 1 2 3 4 5 x

3 6 Complex gamma function The Mittag-Leffler function |!(z)|

One of the most common - and unfounded - reasons as to why Nobel decided against a Nobel prize in math is that [a woman he proposed to/his wife/his mistress] [rejected him because of/cheated him with] a famous mathematician. Gosta Mittag-Leffler is often claimed to be the guilty party. E. Jahnke and F. Emde, There is no historical evidence to support the story. Funktionentafeln mit Formeln und Kurven. Matlab R2007a Leipzig : B.G. Teubner, 1909 In 1882 he founded the Acta Mathematica, which a century later is still one of the world's leading mathematical journals. He persuaded King Oscar II to endow prize competitions and honor various distinguished mathematicians all http://www.mathworks.com/company/newsletters/news_notes/clevescorner/winter02_cleve.html over Europe. Hermite, Bertrand, Weierstrass, and Poincare were among those honored by the King. (http://db.uwaterloo.ca/~alopez-o/math-faq/node50.html) 7 10

Euler’s Beta function The Mittag-Leffler function

Humbert, P. and Agarwal, R.P. (1953). Sur la fonction de Mittag-Leffler et quelques-unes de ses generalisations, Bull. Sci. Math. (Ser. II) 77, 180-185. Let’s establish the relationship between gamma and beta. LT

Inv LT t=1

and in general

8 11

Particular values of !(z) The Mittag-Leffler function can be obtained with the help of B(z,w)

z = "

z = n + "

9 12 The Mittag-Leffler function Laplace transform of MLF

Miller-Ross function: Applied Mathematics and Computation 174 (2006) 869–876 www.elsevier.com/locate/amc After substitutions (which?): Rabotnov function: ! Numerical solution of semidifferential equations by! collocation method

13 16 E.A. Rawashdeh

Department of Mathematics, Yarmouk University, Irbid 21110, Jordan

Abstract

We study the numerical solution of semidifferential equation of order 4 by colloca- tion spline method. We derive a system of equations that characterizing the numerical solution. Some numerical examples are also presented to illustrate our results. Laplace transform of ÓLa2005placeElsevier Inc. transfAll rightsormreserved. of MLF

Consider Keywords:ImporFractionaltant diparfferentiticularal equation; case: #Spline = $ =space; " Collocation method

1. Introduction Therefore, UsefulA fractional for solvingdi ffsoerent calledial semidiffequationerentialof the equations:form n=2 n 1 =2 0 D A Dð À Þ A D y t f t Differentiate k times with respect to z: ½ þ 1 þ Á Á Á þ n Š ð Þ ¼ ð Þ is called a semidifferential equation of order n. In this paper, we study the numerical solution of a semidifferential equation of order 4, 14 17

E-mail address: [email protected]

0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.05.029

Laplace transform of Derivatives of MLF

After substitutions (which?):

Take and integer

!

This is a well known Laplace pair.

15 18 Derivatives of MLF Integration of MLF

We have:

Take Term-by-term integration gives:

Setting and using

19 22

Differential equations for MLF Integration of MLF

Denote:

Then these functions satisfy the following equations:

20 23

Summation formulas for MLF The Wright function

Particular cases:

Replacing # with #/m and with

Taking and gives The Wright function is a generalization of the exponential function and the Bessel functions.

21 24 Laplace transform of the W function

The Mittag-Leffler function!

25

The Mainardi function

The M function is a particular case of the W function:

26