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Laplace-transform formula for a semigroup resolvent
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Mathematics
Area of mathematics
Analysis
Functional analysis
C0-semigroup
Hille-Yosida theorem
2026-09-28
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If
A
generates
a
C0-semigroup
satisfying
∥
U
(
t
)
∥
≤
M
e
ω
t
, then for
Re
z
>
ω
its
resolvent operator
is
(
z
I
−
A
)
−
1
x
=
∫
0
∞
e
−
z
t
U
(
t
)
x
d
t
.
(1)
Repeated differentiation gives
n
!
(
z
I
−
A
)
−
(
n
+
1
)
x
=
∫
0
∞
t
n
e
−
z
t
U
(
t
)
x
d
t
.
Ancestors
(7)
Hille-Yosida theorem
C0-semigroup
Functional analysis
Analysis
Area of mathematics
Mathematics
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Past exam of the mathematics course of the University of Cambridge
/
2022
/
iii
/
Paper 319
/
1
/
a
/
Solution
Past exam of the mathematics course of the University of Cambridge
/
2022
/
iii
/
Paper 319
/
1
/
b
/
Solution
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