Inverse Laplace Transforms

The Inverse Laplace Transform, denoted by $mathcal{L}^{-1}{F(s)}$, converts a function of the complex variable $s$ back to a function of the real variable $t$. It is used to find the time-domain solution of systems whose behavior is described by Laplace Transforms, especially for solving differential equations.

Unlike the forward transform, the inverse transform is often found by recognizing common patterns in $F(s)$ and using a table of Laplace Transforms in reverse, or by decomposing complex $F(s)$ functions into simpler forms (e.g., using partial fraction decomposition) whose inverse transforms are known.

The Laplace Transform is an integral transform that converts a function of a real variable $t$ (often time) to a function of a complex variable $s$ (complex frequency). It is a powerful tool for solving linear ordinary and partial differential equations, particularly in engineering and physics.

Understanding the properties of Laplace Transforms is crucial for effectively applying them to solve problems.

$f(t)$	$F(s) = mathcal{L}{f(t)}$	Conditions
$1$ or $u(t)$	$frac{1}{s}$	$s > 0$
$t$	$frac{1}{s^2}$	$s > 0$
$t^n$ ($n=$ positive integer)	$frac{n!}{s^{n+1}}$	$s > 0$
$t^a$ ($a > -1$, real)	$frac{Gamma(a+1)}{s^{a+1}}$	$s > 0$
$e^{at}$	$frac{1}{s-a}$	$s > a$
$sin(at)$	$frac{a}{s^2+a^2}$	$s > 0$
$cos(at)$	$frac{s}{s^2+a^2}$	$s > 0$
$t e^{at}$	$frac{1}{(s-a)^2}$	$s > a$
$t^n e^{at}$	$frac{n!}{(s-a)^{n+1}}$	$s > a$
$e^{at}sin(bt)$	$frac{b}{(s-a)^2+b^2}$	$s > a$
$e^{at}cos(bt)$	$frac{s-a}{(s-a)^2+b^2}$	$s > a$
$sinh(at)$	$frac{a}{s^2-a^2}$	$s > |a|$
$cosh(at)$	$frac{s}{s^2-a^2}$	$s > |a|$
$delta(t)$ (Dirac delta)	$1$
$u(t-c)$ (Heaviside step)	$frac{e^{-cs}}{s}$	$c ge 0, s > 0$

The First Shifting Theorem states that if $mathcal{L}{f(t)} = F(s)$, then the Laplace Transform of $e^{at}f(t)$ is $F(s-a)$.

Derivation:

This property is very useful for finding transforms of functions that are products of an exponential term and another function whose transform is known.

This property relates the Laplace Transform of the derivative of a function to the transform of the function itself and its initial values. It is key to solving differential equations.

Transform of the First Derivative, $f'(t)$:

Where $f(0)$ is the initial value of $f(t)$ at $t=0$.

Derivation (using integration by parts $int u dv = uv - int v du$):

Transform of the Second Derivative, $f''(t)$:

Where $f'(0)$ is the initial value of the first derivative at $t=0$.

Derivation (applying the first derivative rule twice):

This pattern can be generalized for higher-order derivatives.

Heaviside Unit Step Function, $u(t-c)$

The Heaviside Unit Step Function, often denoted as $u(t)$, $H(t)$, or $theta(t)$, is a discontinuous function whose value is zero for negative arguments and one for positive arguments.

Where $c ge 0$ is the time at which the step occurs. It is fundamental for representing signals that are "switched on" at a specific time, and for defining piecewise functions.

Laplace Transform of the Heaviside Function:

Derivation:

Second Shifting Theorem (Time Shifting)

The Second Shifting Theorem, also known as the Time-Shifting Theorem, states that if $mathcal{L}{f(t)} = F(s)$, then the Laplace Transform of $f(t-c)u(t-c)$ is $e^{-cs}F(s)$.

Where $u(t-c)$ is the Heaviside unit step function, and $c ge 0$. This theorem describes how a delay in the time domain corresponds to multiplication by an exponential term in the s-domain.

Derivation:

This theorem is crucial for analyzing systems with delayed inputs or for converting piecewise functions into the s-domain.

Initial Value Theorem (IVT)

The Initial Value Theorem allows us to find the initial value of a function $f(t)$ (i.e., $f(0^+) = lim_{t to 0^+} f(t)$) directly from its Laplace Transform $F(s)$, without needing to find the inverse transform.

Conditions: This theorem applies if $f(t)$ and $f'(t)$ are Laplace transformable, and the limit $lim_{s to infty} sF(s)$ exists.

It is useful for checking the initial conditions in the solution of differential equations or analyzing system behavior at $t=0$. The theorem is derived from the Laplace transform of a derivative: $mathcal{L}{f'(t)} = int_0^infty e^{-st}f'(t)dt = sF(s) - f(0^+)$. As $s to infty$, the integral $int_0^infty e^{-st}f'(t)dt to 0$ (if $f'(t)$ is bounded or of exponential order). Thus, $0 approx lim_{s to infty} sF(s) - f(0^+)$, leading to the theorem.

Final Value Theorem (FVT)

The Final Value Theorem allows us to find the final (steady-state) value of a function $f(t)$ (i.e., $lim_{t to infty} f(t)$) directly from its Laplace Transform $F(s)$.

Conditions for Applicability: This theorem is only valid if the limit on the right exists AND all poles of $sF(s)$ are in the left-half of the s-plane (i.e., have negative real parts). If $sF(s)$ has any poles on the imaginary axis (except possibly a single pole at the origin $s=0$) or in the right-half plane, the FVT does not apply, and $f(t)$ may not settle to a finite steady-state value.

It is useful for determining the steady-state response of systems. The derivation also starts from $mathcal{L}{f'(t)}$. As $s to 0$, $int_0^infty e^{-st}f'(t)dt to int_0^infty f'(t)dt = [f(t)]_0^infty = lim_{t to infty}f(t) - f(0^+)$. Thus, $lim_{t to infty}f(t) - f(0^+) = lim_{s to 0} [sF(s) - f(0^+)]$, which simplifies to the theorem.