How to Compute Laplace Transform: Simple Steps and Example

To compute the Laplace transform of a function f(t), use the integral formula L{f(t)} = ∫₀^∞ e^{-st} f(t) dt, where s is a complex variable. This transforms the time-domain function into the complex frequency domain, simplifying analysis of signals and systems.

Syntax

The Laplace transform of a function f(t) is defined as:

L{f(t)} = ∫₀^∞ e^{-st} f(t) dt

• f(t): The original function in time domain, defined for t ≥ 0.
• s: A complex number variable s = σ + jω, where σ and ω are real numbers.
• e^{-st}: The kernel that weights the function f(t) during integration.
• ∫₀^∞: The integral from zero to infinity, capturing the entire time domain.

This integral transforms f(t) from the time domain to the complex frequency domain.

latex

L{f(t)} = \int_0^\infty e^{-st} f(t) \, dt

Example

This example computes the Laplace transform of the function f(t) = e^{2t}. The result is F(s) = 1 / (s - 2) for s > 2.

python

import sympy as sp

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

# Define the function f(t)
f = sp.exp(2*t)

# Compute Laplace transform
F = sp.laplace_transform(f, t, s, noconds=True)

print(F)

Output

1/(s - 2)

Common Pitfalls

• Forgetting the integral limits from 0 to infinity instead of negative infinity.
• Mixing up the variable t (time) and s (complex frequency).
• Ignoring the region of convergence where the transform is valid.
• Not using the exponential kernel e^{-st} correctly in the integral.

Always check the function domain and convergence conditions.

python

import sympy as sp

t, s = sp.symbols('t s', real=True)

# Wrong: Using integral from -inf to inf
f = sp.exp(2*t)
wrong_integral = sp.integrate(sp.exp(-s*t)*f, (t, -sp.oo, sp.oo))

# Right: Integral from 0 to inf
right_integral = sp.integrate(sp.exp(-s*t)*f, (t, 0, sp.oo))

print('Wrong integral:', wrong_integral)
print('Right integral:', right_integral)  # May diverge if Re(s) <= 2

Output

Wrong integral: oo
Right integral: 1/(s - 2)

Quick Reference

Function f(t)	Laplace Transform F(s)	Region of Convergence
1	1/s	Re(s) > 0
t	1/s^2	Re(s) > 0
e^{at}	1/(s - a)	Re(s) > a
sin(bt)	b/(s^2 + b^2)	Re(s) > 0
cos(bt)	s/(s^2 + b^2)	Re(s) > 0

Key Takeaways

The Laplace transform converts time functions into complex frequency domain using an integral from 0 to infinity.

Use the formula L{f(t)} = ∫₀^∞ e^{-st} f(t) dt with s as a complex variable.

Check the region of convergence to ensure the transform is valid.

Common mistakes include wrong integral limits and confusing variables t and s.

Symbolic tools like SymPy can compute Laplace transforms easily and correctly.