Laplace Transforms Classroom

95.1 Introduction to Laplace Transforms

The Laplace Transform is a powerful mathematical tool used to convert a function of a real variable $t$ (often time) to a function of a complex variable $s$ (complex frequency). It simplifies the process of solving linear differential equations, especially those with initial conditions.

• Purpose:
  1. Transforms differential equations into algebraic equations.
  2. Simplifies the solution of initial value problems.
  3. Widely used in engineering (control systems, circuit analysis) and physics.
• Key Idea: Converts operations in the time domain (differentiation, integration) into simpler operations (multiplication, division) in the s-domain.

95.2 Definition of Laplace Transform

The Laplace Transform of a function $f(t)$, denoted as $\mathcal{L}\{f(t)\}$ or $F(s)$, is defined by the integral:

where $s$ is a complex variable ($\sigma + i\omega$), and the integral converges for $\text{Re}(s) > a$ for some constant $a$.

• $f(t)$: The function in the time domain (usually $t \ge 0$).
• $F(s)$: The transformed function in the s-domain.
• $e^{-st}$: The kernel of the transform.

95.3 Linearity Property

The Laplace Transform is a linear operator. This means that the transform of a sum of functions is the sum of their transforms, and the transform of a constant times a function is the constant times the transform of the function.

Mathematically, for constants $a$ and $b$, and functions $f(t)$ and $g(t)$:

This property is extremely useful as it allows us to break down complex functions into simpler parts, transform each part, and then combine the results.

95.4 Elementary Function Transforms

Here are some common Laplace Transforms of elementary functions:

• $\mathcal{L}\{1\} = \frac{1}{s}$
• $\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$ (for $n=0, 1, 2, \dots$)
• $\mathcal{L}\{e^{at}\} = \frac{1}{s-a}$
• $\mathcal{L}\{\sin(at)\} = \frac{a}{s^2 + a^2}$
• $\mathcal{L}\{\cos(at)\} = \frac{s}{s^2 + a^2}$

These basic transforms form the building blocks for transforming more complex functions using properties like linearity.

95.5 Standard Transform Problems

Applying the definition and elementary transforms to find the Laplace Transform of various functions.

96.1 Transform of $e^{at}f(t)$ (First Shifting Theorem)

The First Shifting Theorem, also known as the frequency shifting property, states that if $\mathcal{L}\{f(t)\} = F(s)$, then the Laplace Transform of $e^{at}f(t)$ is:

This means that multiplying a function $f(t)$ by $e^{at}$ in the time domain corresponds to shifting the argument of its Laplace Transform $F(s)$ by $a$ in the s-domain.

96.2 Transforms of $e^{at}f(t)$ (Examples)

Let's apply the First Shifting Theorem to some examples.

96.3 Transforms of Derivatives

The Laplace Transform of derivatives is crucial for solving differential equations. It converts differentiation in the time domain to multiplication by $s$ in the s-domain, plus terms involving initial conditions.

• First Derivative: $$\mathcal{L}\{f'(t)\} = sF(s) - f(0)$$
• Second Derivative: $$\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f'(0)$$
• General $n$-th Derivative: $$\mathcal{L}\{f^{(n)}(t)\} = s^nF(s) - s^{n-1}f(0) - s^{n-2}f'(0) - \dots - f^{(n-1)}(0)$$