Initial value theorem
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 be the (one-sided) Laplace transform of "ƒ"("t"). If formula_2 is bounded on formula_3 (or if just formula_4) and formula_5 exists then the initial value theorem says Proof. Suppose first that formula_7 is bounded. Say formula_8. A change of variable in the integral formula_9 shows that Since formula_2 is bounded, the Dominated Convergence Theorem shows that Of course we don't really need  DCT here, one can give a very simple proof using only elementary calculus: Start by choosing formula_13 so that formula_14, and then note that formula_15 "uniformly" for formula_16. The theorem assuming just that formula_4 follows from the theorem for bounded formula_2: Define formula_19. Then formula_20 is bounded, so we've shown that formula_21. But formula_22 and formula_23, so since formula_25