Page 126 - Mathematical Techniques of Fractional Order Systems

114 Mathematical Techniques of Fractional Order Systems

Lemma 6: Define the lower triangular matrix
0 0 0 ? 0 1
e 1;1
0 ?
e 2;1 e 2;2
B 0 C k
i
B ? 0 C; e k;i 5 21Þ Γ ðk2iÞ
C
e 3;1 e 3;2 e 3;3 ð 1 ðÞ ð4:55Þ
E 5 B
^ ^ ^ & ^
@ A i
?
e N;1 e N;2 e N;3 e N;N
in which Γ ðk2iÞ 1 ðÞ denotes the k 2 iÞ-th derivative of Gamma function at
ð
point 1 and N $ k is any natural number. We have
k
X
i k
ðiÞ
1
b k;i L t-s ln t 2 Γ ðÞ=s 5 ln sðÞ=s ð4:56Þ
i51

where b k;i are the elements of the matrix E 21 5 b k;i .
Proof: Consider the Laplace transform of the power function as
ð 1N Γ a 1 1Þ
ð
a 2st
t e dt 5 a11 ; a .2 1 ð4:57Þ
0 s
Inspired by Pahikkala (2013), we differentiate (4.57) with respect to a,
which gives
1N
ð1Þ
ð Γ ð a 1 1Þ 2 lnsΓ a 1 1Þ
ð
a
t lnte 2st dt 5 a11 ð4:58Þ
0 s
A mathematical induction indicates that for the k-th derivative one would
have
1N k
ð 1 X k
k 2st
a
t lntÞ e dt 5 a11 Γ ð a 1 1Þ 2lnsÞ k2i ð4:59Þ
ðiÞ
ð
ð
0 s i50 i
In order to prove this, assume (4.59) holds true for some k . 0.
Differentiating both sides of (4.59) with respect to a yields
s a11 Ð 1N a ð k11 2st dt 5
e
t lntÞ
0
k k
X k Γ ði11Þ k2i 1 X k ðiÞ k2i11
ð
ð
i ð a 1 1Þ 2lnsÞ i Γ ð a 1 1Þ 2lnsÞ
i50 i50
k11
X k k2i11
5 Γ ð a 1 1Þ 2lnsÞ 1
ðiÞ
ð
i 2 1
i51
k ð4:60Þ
X k k2i11
ðiÞ
ð
Γ ð a 1 1Þ 2lnsÞ
i
i50
5 Γ ðk11Þ ð a 1 1Þ 1
k
X k 1 k ðiÞ k2i11 k11
ð
ð
ð
i 2 1 i Γ ð a 1 1Þ 2lnsÞ 1 Γ a 1 1Þ 2lnsÞ
i51