Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

3.4 Generalized first-order kinetics* 54
(1) Laplace transforms provide a convenient method for solving differential equations,
(2) Laplace transforms form the connection between microscopic molecular properties
and statistically (Boltzmann) averaged quantities.
I
Definition 3.2: The Laplace transform L [ ()] of a function () is defined as the
integral
Z ∞
() =L [ ()] = () − (3.117)
where
0
• is a real variable,
• () is a real function of the variable with the property () =0for 0,
• is a complex variable,
• () =L [ ()] is a function of the variable .
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Definition 3.3: The inverse Laplace transform L −1 [ ()] of the function () is
defined as the integral
() =L −1 [ ()] = 1 Z
2
+∞
−∞
() (3.118)
where
• is an arbitrary real constant.
I Relation between f (t) and F (p): Since L −1 [ ()] recovers the original function
(), the pair of functions () and () is said to form a Laplace pair:
L −1 [ ()] = L −1 [L [ ()]] = () (3.119)
and
L [ ()] = L £ L −1 [ ()] ¤ = () (3.120)
A list of Laplace and inverse Laplace transforms of some functions can be found in
Appendix E (Table E.1).