In mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.
It is also known under the abbreviation IVT.
Let
F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t be the (one-sided) Laplace transform of ƒ(t). The initial value theorem then says
lim t → 0 f ( t ) = lim s → ∞ s F ( s ) . Based on the definition of Laplace transform of derivative we have:
s F ( s ) = f ( 0 − ) + ∫ t = 0 − ∞ e − s t f ′ ( t ) d t thus:
lim s → ∞ s F ( s ) = lim s → ∞ [ f ( 0 − ) + ∫ t = 0 − ∞ e − s t f ′ ( t ) d t ] But lim s → ∞ e − s t is indeterminate between t=0− to t=0+; to avoid this, the integration can be performed in two intervals:
lim s → ∞ [ ∫ t = 0 − ∞ e − s t f ′ ( t ) d t ] = lim s → ∞ { lim ϵ → 0 + [ ∫ t = 0 − ϵ e − s t f ′ ( t ) d t ] + lim ϵ → 0 + [ ∫ t = ϵ ∞ e − s t f ′ ( t ) d t ] } In the first expression,
e − s t = 1 w h e n 0 − < t < 0 + . In the second expression, the order of integration and limit-taking can be changed. Also
lim s → ∞ e − s t ( t ) = 0 w h e r e 0 + < t < ∞ . Therefore:
lim s → ∞ [ ∫ t = 0 − ∞ e − s t f ′ ( t ) d t ] = lim s → ∞ { lim ϵ → 0 + [ ∫ t = 0 − ϵ f ′ ( t ) d t ] } + lim ϵ → 0 + { ∫ t = ϵ ∞ lim s → ∞ [ e − s t f ′ ( t ) d t ] } = f ( t ) | t = 0 − t = 0 + + 0 = f ( 0 + ) − f ( 0 − ) + 0 By substitution of this result in the main equation we get:
lim s → ∞ s F ( s ) = f ( 0 − ) + f ( 0 + ) − f ( 0 − ) = f ( 0 + ) 