Question

How do you write a Laplace transform?

Answer 1

The Laplace transform is a powerful tool in engineering and mathematics, especially for solving differential equations and analyzing linear time-invariant systems. Here's a step-by-step guide on how to write and compute a Laplace transform:

### Definition

The Laplace transform \( \mathcal{L} \) of a function \( f(t) \) is defined by the following integral:

\[ \mathcal{L}\{f(t)\} = F(s) = \int_0^\infty f(t) e^{-st} \, dt \]

where:
- \( t \) is the time variable,
- \( s \) is a complex number frequency parameter (\( s = \sigma + j\omega \)),
- \( f(t) \) is the time-domain function,
- \( F(s) \) is the transformed function in the Laplace domain.

### Steps to Compute the Laplace Transform

1. **Identify the Function \( f(t) \)**:
   - Determine the function you want to transform. This function should be defined for \( t \geq 0 \).

2. **Set Up the Integral**:
   - Substitute \( f(t) \) into the Laplace transform formula:
     \[
     \mathcal{L}\{f(t)\} = \int_0^\infty f(t) e^{-st} \, dt
     \]

3. **Evaluate the Integral**:
   - Compute the integral to find \( F(s) \). This often involves integration techniques and might require some manipulation depending on the complexity of \( f(t) \).

4. **Simplify the Result**:
   - Simplify \( F(s) \) to get it in a standard or more usable form.

### Examples

1. **Example 1: Exponential Function**

   For \( f(t) = e^{at} \):
   \[
   \mathcal{L}\{e^{at}\} = \int_0^\infty e^{at} e^{-st} \, dt = \int_0^\infty e^{(a-s)t} \, dt
   \]
   To solve this:
   \[
   \int_0^\infty e^{(a-s)t} \, dt = \left[ \frac{e^{(a-s)t}}{a-s} \right]_0^\infty
   \]
   Assuming \( \text{Re}(s) > \text{Re}(a) \), this evaluates to:
   \[
   \mathcal{L}\{e^{at}\} = \frac{1}{s-a}
   \]

2. **Example 2: Sine Function**

   For \( f(t) = \sin(bt) \):
   \[
   \mathcal{L}\{\sin(bt)\} = \int_0^\infty \sin(bt) e^{-st} \, dt
   \]
   To solve this:
   \[
   \int_0^\infty \sin(bt) e^{-st} \, dt = \frac{b}{s^2 + b^2}
   \]
   Thus:
   \[
   \mathcal{L}\{\sin(bt)\} = \frac{b}{s^2 + b^2}
   \]

### Common Laplace Transforms

- **Constant Function \( f(t) = 1 \)**:
  \[
  \mathcal{L}\{1\} = \frac{1}{s}
  \]

- **Unit Step Function \( f(t) = u(t) \)**:
  \[
  \mathcal{L}\{u(t)\} = \frac{1}{s}
  \]

- **Delta Function \( f(t) = \delta(t) \)**:
  \[
  \mathcal{L}\{\delta(t)\} = 1
  \]

- **Derivative of a Function**:
  If \( \mathcal{L}\{f(t)\} = F(s) \), then:
  \[
  \mathcal{L}\{f'(t)\} = sF(s) - f(0)
  \]

Understanding and using the Laplace transform can greatly simplify the process of solving differential equations and analyzing system behaviors.

Answer 2

Writing a Laplace transform involves transforming a function of time, \( f(t) \), into a function of a complex variable, \( s \). This process is useful in various fields such as engineering, physics, and applied mathematics, particularly for solving differential equations.

Here's a detailed breakdown of how to write and compute a Laplace transform:

### 1. **Understanding the Definition**

The Laplace transform \( \mathcal{L}\{f(t)\} \) of a function \( f(t) \) is defined by the integral:

\[ \mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt \]

where:
- \( t \) is the time variable (usually \( t \geq 0 \)),
- \( s \) is a complex variable (usually \( s = \sigma + j\omega \), where \( \sigma \) and \( \omega \) are real numbers, and \( j \) is the imaginary unit),
- \( F(s) \) is the resulting function in the \( s \)-domain.

### 2. **Applying the Definition**

To apply the definition, follow these steps:

- **Identify the function \( f(t) \)**: This is the time-domain function you want to transform.
- **Set up the integral**: Substitute \( f(t) \) into the integral formula \( \int_{0}^{\infty} e^{-st} f(t) \, dt \).
- **Evaluate the integral**: Perform the integration with respect to \( t \).

### 3. **Example**

Let’s compute the Laplace transform of a simple function, \( f(t) = e^{at} \), where \( a \) is a constant.

Using the definition:

\[ \mathcal{L}\{e^{at}\} = \int_{0}^{\infty} e^{-st} e^{at} \, dt \]

Combine the exponents:

\[ = \int_{0}^{\infty} e^{(a - s)t} \, dt \]

This is an exponential integral. For convergence, \( \text{Re}(s) > a \). Integrate:

\[ \int_{0}^{\infty} e^{(a - s)t} \, dt = \left[ \frac{e^{(a - s)t}}{a - s} \right]_{0}^{\infty} \]

Evaluating the integral:

- At \( t = \infty \), \( e^{(a - s)t} \) approaches 0 if \( \text{Re}(s) > a \).
- At \( t = 0 \), \( e^{(a - s)t} = 1 \).

Thus:

\[ \mathcal{L}\{e^{at}\} = \frac{1}{s - a} \]

### 4. **Common Laplace Transforms**

Here are some standard transforms you might find useful:

- **Constant function** \( f(t) = 1 \): \( \mathcal{L}\{1\} = \frac{1}{s} \)
- **Power function** \( f(t) = t^n \): \( \mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}} \)
- **Sine function** \( f(t) = \sin(\omega t) \): \( \mathcal{L}\{\sin(\omega t)\} = \frac{\omega}{s^2 + \omega^2} \)
- **Cosine function** \( f(t) = \cos(\omega t) \): \( \mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2 + \omega^2} \)

### 5. **Inverse Laplace Transform**

To find the original function \( f(t) \) from \( F(s) \), you use the inverse Laplace transform, denoted \( \mathcal{L}^{-1}\{F(s)\} \). The inverse transform is not always straightforward but follows similar principles as the Laplace transform.

### Summary

To write and compute a Laplace transform, you:
1. Identify the time-domain function \( f(t) \).
2. Set up and evaluate the integral \( \int_{0}^{\infty} e^{-st} f(t) \, dt \).
3. Simplify the result to find \( F(s) \).

Understanding and applying these steps allows you to transform functions between the time domain and the complex \( s \)-domain, which is particularly helpful for solving linear differential equations and analyzing systems.