Math 2.0

Integral transforms, operational calculus Laplace transform: operational rules Theorem

Initial value theorem

Builds on Laplace transformLimit of a function

Statement

Let f ⁣:[0,∞)→ℂf \colon [0, \infty) \to \C be integrable on each interval [0,T][0, T]. Assume that the Laplace integral of ff converges at some real number s0s_0, and let FF be the Laplace transform of ff. Assume that the one-sided limit f(0+)=lim⁡t→0+f(t)f(0^+) = \lim_{t \to 0^+} f(t) exists and is finite. Then lim⁡s→+∞sF(s)=f(0+),\lim_{s \to +\infty} s F(s) = f(0^+), where ss tends to +∞+\infty through real values.

Remarks

Since s∫0∞e−st dt=1s \int_0^\infty e^{-s t} \, dt = 1, the difference sF(s)−f(0+)s F(s) - f(0^+) is s∫0∞e−st(f(t)−f(0+)) dts \int_0^\infty e^{-s t} (f(t) - f(0^+)) \, dt. The weight se−sts e^{-s t} has total mass 11 and concentrates near t=0t = 0 as s→∞s \to \infty. On [0,δ][0, \delta] the integrand is small by the existence of f(0+)f(0^+); on [δ,∞)[\delta, \infty) the contribution tends to 00 by the convergence at s0s_0. Example: for f(t)=cos⁡tf(t) = \cos t, sF(s)=s2/(s2+1)→1=f(0)s F(s) = s^2 / (s^2 + 1) \to 1 = f(0). The finiteness of f(0+)f(0^+) matters: for f(t)=t−1/2f(t) = t^{-1/2} one has F(s)=π/sF(s) = \sqrt{\pi / s}, so sF(s)=πs→+∞s F(s) = \sqrt{\pi s} \to +\infty.

Going deeper

For a student

In control and circuit theory one uses it to check the value of a response just after a switch is closed. It is the simplest Abelian theorem for the Laplace transform: behaviour of ff near 00 gives behaviour of FF near ∞\infty. Its companion is the final value theorem, which links ff near ∞\infty with FF near 00.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Function, concept, no evidence levelFunctionconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionLaplace transform, definition, no evidence levelLaplace transformdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionInitial value theorem, statement, no evidence levelInitial value theoremstatement
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  1. Function concept
  2. Limit point of a set definition
  3. Laplace transform definition
  4. Limit of a function definition
  5. Initial value theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Initial_value_theorem
About this page
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Kindstatement, theorem
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AboutLaplace transform
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