Initial value theorem

The initial value theorem reads off a signal’s value at $t=0^{+}$ straight from its Laplace transform, no inversion needed:

$x(0^{+})={\lim }_{s\to \infty }sX(s),$

provided $x(t)$ has no impulses at $t=0$.

Why it works

For a unilateral transform $X(s)={\int }_0^{\infty }x(t)e^{-st}dt$, as $s\to \infty$ the kernel $e^{-st}$ concentrates more and more sharply at $t=0^{+}$, so the integral $sX(s)$ approaches the initial value of $x$. The factor of $s$ scales the kernel back to integral $1$ in the limit.

A non-rigorous but quick derivation: by the unilateral differentiation property, $\mathcal{L}\{x^{\prime }(t)\}=sX(s)-x(0^{-})$. Take $s\to \infty$ on both sides. The left side ${\int }_0^{\infty }{x}^{\prime }(t)e^{-st}dt$ goes to zero (the kernel vanishes for any fixed $t>0$, leaving only a small contribution near $t=0$ from the jump $x(0^{+})-x(0^{-})$). The right side becomes $\lim sX(s)-x(0^{-})$. Equating: $\lim sX(s)=x(0^{-})+(x(0^{+})-x(0^{-}))=x(0^{+})$.

Worked example

The IVT requires $X(s)$ to be strictly proper (numerator degree strictly less than denominator degree). Try it on something improper like $X(s)=(s+2)/(s+4)$, same degrees, and you get

${\lim }_{s\to \infty }sX(s)={\lim }_{s\to \infty }\frac{s^2+2s}{s+4}=\infty ,$

which diverges. That’s the signature that $x(t)$ contains an impulse at $t=0$ and the IVT doesn’t apply.

For a proper $X(s)$ like $X(s)=4/(s+3)$:

${\lim }_{s\to \infty }\frac{4s}{s+3}=4.$

So $x(0^{+})=4$. Check: $x(t)=4e^{-3t}u(t)$ gives $x(0^{+})=4$. ✓

When it’s useful

You’re given $X(s)$ as a rational function and want the initial value of $x(t)$ without inverting. Comes up checking solutions to ODE initial-value problems, or characterizing step responses without computing the full time-domain function.

Conditions

• $x(t)$ must have no impulse at $t=0$ (so $X(s)$ must be a proper rational function).
• Either $x(0^{+})$ exists, or $X(s)$ converges in some right half-plane.

For most engineering problems the conditions hold, so the IVT works as a fast sanity check.

See also

The companion result for long-time behavior is the Final value theorem: ${\lim }_{t\to \infty }x(t)={\lim }_{s\to 0}sX(s)$.