This paper reviews some properties of the gamma function, particularly the incomplete gamma function and its complement, as a function of the Laplace variable $s$. The utility of these functions in the solution of initialization problems in fractional-order system theory is demonstrated. Several specific differential equations are presented, and their initialization responses are found for a variety of initializations. Both the time-domain and Laplace-domain solutions are obtained and compared. The complementary incomplete gamma function is shown to be essential in finding the Laplace-domain solution of a fractional-order differential equation.

1.
Podlubny
,
I.
, 1999,
Fractional Differential Equations
,
,
New York
.
2.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
, 2007, “
Initialization of Fractional Differential Equations: Theory and Application
,”
Proceedings of IDETC∕CIE 2007, ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Sept. 4–7, Las Vegas, NV, Paper No. DETC2007–34814.
3.
Spanier
,
J.
, and
Oldham
,
K. B.
, 1987,
An Atlas of Functions
,
Hemisphere
,
New York
.
4.
Roberts
,
G. E.
, and
Kaufman
,
H.
, 1966,
Table of Laplace Transforms
,
W. B.
Saunders Co.
,
.
5.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
, 2007, “
Initialization of Fractional Differential Equations: Background and Theory
,”
Proceedings of IDETC∕CIE 2007, ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Sept. 4–7, Las Vegas, NV, Paper No. DETC2007-34814.