Biomedical Engineering Reference
In-Depth Information
Appendix
Useful Notes on Fractional Calculus
In this appendix, some useful concepts from fractional calculus will be presented for
the reader. For more details, the reader is encouraged to read more comprehensive
works on this topic [ 105 , 118 , 126 , 129 , 146 , 147 , 149 , 150 ].
We will start by introducing some basic functions. The gamma function is in-
trinsically tied to fractional calculus by definition. The simplest interpretation of
the gamma function is the generalization of the factorial for all real numbers. The
definition of the gamma function is given by
e u u z 1 du,
(z)
=
for all z
R
(A.1)
0
The 'beauty' of the gamma function can be found in its properties:
(z
+
1 )
=
z(z)
(A.2)
(z)
=
(z
1 )
!
The consequence of this relation for integer values of z is the definition for factorial.
Using the gamma function we can also define the function Φ(t) , which later will
become useful for showing alternate forms of the fractional integral:
t α 1
+ (α)
φ α (t)
=
(A.3)
Also known as the Euler Integral of the First Kind, the Beta Function is in impor-
tant relationship in fractional calculus. Its solution not is only defined through the
use of multiple Gamma Functions, but furthermore shares a form that is character-
istically similar to the Fractional Integral/Derivative of many functions, particularly
polynomials of the form t α
and the Mittag-Leffer Function:
1
u) p 1 uq
B(p,q) =
( 1
1 du
0
(p)(q)
(p + q) = B(q,p),
=
where p,q R +
(A.4)
 
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