Laplace & Integral Transforms

The Laplace transform is a powerful integral transform that maps functions from the time domain ($t$) to the complex frequency domain ($s$). In university-level analysis and engineering, it serves as the primary tool for solving linear ordinary differential equations (ODEs), particularly those involving discontinuous forcing functions or impulsive inputs where traditional methods like undetermined coefficients or variation of parameters become cumbersome.

1. Formal Definition and the Region of Convergence (ROC)

Let $f(t)$ be a function defined for $t\ge 0$. The unilateral Laplace transform of $f(t)$, denoted by $\mathcal{L}\{f(t)\}$ or $F(s)$, is defined by the improper integral:

$F(s)=\mathcal{L}\{f(t)\}(s)={\int }_0^{\infty }{e}^{-st}f(t)dt$

where $s=\sigma +i\omega$ is a complex variable.

The Region of Convergence (ROC)

The integral exists only if it converges. For a function $f(t)$ to have a Laplace transform, it must be piecewise continuous on every finite interval in $[0,\infty )$ and of exponential order. A function is of exponential order $c$ if there exist constants $M>0$ and $T>0$ such that $|f(t)|\le M{e}^{ct}$ for all $t>T$.

If $f(t)$ is of exponential order $c$, then the integral converges for $\text{Re}(s)>c$. This half-plane in the complex $s$-domain is known as the Region of Convergence (ROC).

2. Fundamental Properties

The utility of the Laplace transform stems from its operational properties, which allow us to manipulate calculus problems algebraically.

Linearity

The Laplace transform is a linear operator. For constants $a,b$ and functions $f(t),g(t)$: $\mathcal{L}\{af(t)+bg(t)\}=aF(s)+bG(s)$

Scaling

If $\mathcal{L}\{f(t)\}=F(s)$, then for $a>0$: $\mathcal{L}\{f(at)\}=\frac{1}{a}f\left(\frac{s}{a}\right)$

First Shifting Theorem (Frequency Shifting)

$\mathcal{L}\{e^{at}f(t)\}=F(s-a)$

3. Differentiation and Integration in the s-domain

The “core magic” of the Laplace transform lies in how it handles derivatives.

Differentiation of the Original Function

Given $f(t)$ is continuous and $f^{\prime }(t)$ is piecewise continuous: $\mathcal{L}\{f^{\prime }(t)\}=sF(s)-f(0)$ $\mathcal{L}\{f^{\prime \prime }(t)\}=s^{2}F(s)-sf(0)-f^{\prime }(0)$

In general, for the $n$-th derivative: $\mathcal{L}\{f^{(n)}(t)\}=s^{n}F(s)-{\sum }_{k=1}^n{s}^{n-k}{f}^{(k-1)}(0)$

Integration of the Original Function

$\mathcal{L}\left\{{\int }_0^{t}f(\tau )d\tau \right\}=\frac{F(s)}{s}$

4. Solving Initial Value Problems (IVPs)

The Laplace transform is uniquely suited for IVPs because the initial conditions are incorporated directly into the transformation process. Unlike the method of undetermined coefficients, we do not need to solve for a general solution and then find the constants.

Consider the second-order linear ODE: $a{y}^{\prime \prime }(t)+b{y}^{\prime }(t)+cy(t)=g(t),y(0)=y_0,y^{\prime }(0)=y_1$

Taking the Laplace transform of both sides yields an algebraic equation: $a[s^{2}Y(s)-sy(0)-y^{\prime }(0)]+b[sY(s)-y(0)]+cY(s)=G(s)$

Solving for $Y(s)$: $Y(s)=\frac{G(s)+a(s{y}_0+y_1)+b{y}_0}{a{s}^2+bs+c}$

The solution $y(t)$ is then recovered via the Inverse Laplace Transform ${\mathcal{L}}^{-1}\{Y(s)\}$, often using partial fraction decomposition.

5. Discontinuous Forcing: Step and Impulse Functions

In physical systems, inputs often switch on/off (step) or occur instantaneously (impulse).

Heaviside Step Function $u(t-a)$

Defined as: $u(t-a)=\{\begin{array}{ll}0,& t<a\\ 1,& t\ge a\end{array}$ Its transform is: $\mathcal{L}\{u(t-a)\}=\frac{e^{-as}}{s},s>0$

Second Shifting Theorem: $\mathcal{L}\{f(t-a)u(t-a)\}=e^{-as}F(s)$

Dirac Delta Function $\delta (t-a)$

The Dirac delta is a generalized function (distribution) representing an idealized unit impulse at $t=a$: ${\int }_0^{\infty }\delta (t-a)f(t)dt=f(a)$ Its transform is remarkably simple: $\mathcal{L}\{\delta (t-a)\}=e^{-as}$

6. The Convolution Theorem

The convolution of two functions $f$ and $g$ is defined as: $(f\ast g)(t)={\int }_0^{t}f(\tau )g(t-\tau )d\tau$

The Convolution Theorem states: $\mathcal{L}\{f\ast g\}=F(s)G(s)$

This is vital for finding inverse transforms. If we recognize a transform as a product $F(s)G(s)$, the time-domain solution is the convolution of their respective inverses. This often avoids complex partial fraction decompositions.

7. Transfer Functions and System Dynamics

In control theory, the relationship between input $X(s)$ and output $Y(s)$ of a linear time-invariant (LTI) system is defined by the Transfer Function $H(s)$: $H(s)=\frac{Y(s)}{X(s)}$

If the input is an impulse $\delta (t)$, then $X(s)=1$, and $Y(s)=H(s)$. Thus, the inverse transform $h(t)={\mathcal{L}}^{-1}\{H(s)\}$ is the impulse response of the system.

Stability, Poles, and Zeros

The roots of the denominator of $H(s)$ are the poles, and the roots of the numerator are the zeros.

• A system is stable if all its poles lie in the left half-plane (LHP) of the $s$-domain ($\text{Re}(s)<0$).
• Poles on the imaginary axis correspond to oscillations; poles in the right half-plane indicate exponential divergence (instability).

8. Connection to the Fourier Transform

The Laplace transform can be viewed as a generalization of the Fourier transform. If we set $s=i\omega$ (i.e., $\sigma =0$), the unilateral Laplace transform of a function that is zero for $t<0$ becomes the Fourier transform: $F(i\omega )={\int }_0^{\infty }f(t)e^{-i\omega t}dt$

The $e^{-\sigma t}$ term in the Laplace transform acts as a “convergence factor,” allowing us to transform functions that do not have a Fourier transform (e.g., $f(t)=tu(t)$).

Python Implementation with SymPy

We can use Python’s sympy library to handle symbolic Laplace transforms, inverse transforms, and partial fraction expansions.

import sympy as sp

# Define symbols
t, s = sp.symbols('t s')
a = sp.symbols('a', real=True, positive=True)

# 1. Compute Laplace Transform
f = sp.exp(-a*t) * sp.sin(t)
F = sp.laplace_transform(f, t, s)
print(f"Laplace transform of {f}: {F[0]}")

# 2. Partial Fraction Decomposition
# Let G(s) = (3s + 1) / (s^2 + s)
G_expr = (3*s + 1) / (s**2 + s)
G_pfrac = sp.apart(G_expr)
print(f"Partial fraction of {G_expr}: {G_pfrac}")

# 3. Inverse Laplace Transform
# Recover g(t) from G_pfrac
g = sp.inverse_laplace_transform(G_pfrac, s, t)
print(f"Inverse Laplace: {g}")