Laplace Transform

Yaser Rahmati | یاسر رحمتی

1. Definition:

The Laplace transform of a function $f(t)$, denoted by $L{f(t)}$, is defined as:

$$F(s)=L(f(t))=\int_{0}^{\infty }f(t)e^{-st}dt$$

where $s$ is a complex variable.

2. Transform Pair:

• The Laplace transform is often used to convert differential equations in the time domain into algebraic equations in the Laplace domain, which are easier to solve.

• The transform pair is the relationship between a function $f(t)$ and its Laplace transform $F(s)$.

• For example, if $L{f(t)}=F(s)$, then we say that $f(t)$ and $F(s)$ form a Laplace transform pair.

Example 1

Consider a constant $c$. The function $f(t) = c$ and the following expression is integrated.

$$F(s)=L(f(t))=\int_{0}^{\infty }ce^{-st}dt=-\frac{c}{s}e^{-st}|\begin{matrix} \infty \\ 0 \end{matrix}=0-(-\frac{c}{s})=\frac{c}{s}$$

You can use the sympy library in Python to calculate the Laplace transform of a constant. Here's a Python code snippet to compute it:

import sympy as sp

# Define symbols
s, t, c = sp.symbols('s t c')

# Define the constant function
f = c

# Compute the Laplace transform
laplace_transform = sp.laplace_transform(f, t, s)
laplace_transform_simplified = sp.simplify(laplace_transform[0])

# Print the result
print("Laplace transform of constant c:", laplace_transform_simplified)

Make sure you have the sympy library installed (pip install sympy).

Example 2

Compute $L({e^{at}})$.

Plug the function into the definition of the transform and do a little simplification.

$$L\begin{Bmatrix} e^{at} \end{Bmatrix}=\int_{0}^{\infty }e^{-st}.e^{at}dt=\int_{0}^{\infty }e^{(a-s)t}dt=\frac{1}{s-a}$$

You can use the sympy library in Python to compute the Laplace transform of $e^{at}$. Here's how you can do it:

import sympy as sp

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

# Define the function e^(at)
f = sp.exp(a*t)

# Compute the Laplace transform
laplace_transform = sp.laplace_transform(f, t, s)
laplace_transform_simplified = sp.simplify(laplace_transform[0])

# Print the result
print("Laplace transform of e^(at):", laplace_transform_simplified)

This code will compute and print the Laplace transform of $e^{at}$.

Example 3

The Laplace transform of a sinusoidal function depends on whether it's a sine or cosine function. Here are the Laplace transforms of both:

Laplace transform of sine function:

$$L[sin(ωt)]= \frac{ω}{s^2 +ω^2}$$

Laplace transform of cosine function:

$$L[sin(ωt)]= \frac{s}{s^2 +ω^2}$$

You can use the sympy library in Python to compute these transforms. Here's the code:

import sympy as sp

# Define symbols
s, t, omega = sp.symbols('s t omega')

# Define the sine function
sin_function = sp.sin(omega * t)

# Define the cosine function
cos_function = sp.cos(omega * t)

# Compute the Laplace transforms
laplace_sin = sp.laplace_transform(sin_function, t, s)
laplace_cos = sp.laplace_transform(cos_function, t, s)

# Simplify the transforms
laplace_sin_simplified = sp.simplify(laplace_sin[0])
laplace_cos_simplified = sp.simplify(laplace_cos[0])

# Print the results
print("Laplace transform of sine function:", laplace_sin_simplified)
print("Laplace transform of cosine function:", laplace_cos_simplified)

This code will compute and print the Laplace transforms of both sine and cosine functions.

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