Laplace Transform, Initial and Final Value Theorem

Cuthbert Nyack
Sometimes it may only be necessary to find the behaviour of a function at small and/or large times without finding an explicit expression for the inverse of the Laplace Transform. This is particularly so when the Laplace Transform may be an involved expression. To find the behaviour of f(t) for small times the INITIAL VALUE THEOREM shown below may be used. This implies that the small time behaviour is dominated by high frequencies or poles far from the real s axis.
To find the behaviour of f(t) for large times, the FINAL VALUE THEOREM shown below may be used. This implies that the long time behaviour is determined by low frequencies or poles close to or on the real s axis.

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Copyright 1996 Cuthbert A. Nyack.