Lesson 1: Definition and Basic Transforms
Estimated time: 45-55 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- State the definition of the Laplace transform as an improper integral
- Compute L{1}, L{tn}, L{eat}, L{sin(at)}, and L{cos(at)} from the definition
- Apply the linearity property of the Laplace transform
- Use a table of basic transforms to quickly find L{f(t)}
- State the conditions for existence of the Laplace transform
The Laplace Transform Definition
Laplace Transform: For a function f(t) defined for t ≥ 0, the Laplace transform is L{f(t)} = F(s) = ∫0∞ e-st f(t) dt, provided the integral converges. The variable s is typically a real number (or complex) large enough for convergence.
The Laplace transform converts a function of time t into a function of the frequency variable s. It turns differentiation into multiplication by s, which is why it simplifies differential equations.
Computing Basic Transforms from the Definition
Example 1: L{1}
L{1} = ∫0∞ e-st dt = [-e-st/s]0∞ = 0 - (-1/s) = 1/s, for s > 0.
Example 2: L{eat}
L{eat} = ∫0∞ e-st eat dt = ∫0∞ e-(s-a)t dt = 1/(s-a), for s > a.
Result: L{eat} = 1/(s-a).
Example 3: L{t}
L{t} = ∫0∞ t e-st dt. Using integration by parts (u = t, dv = e-stdt):
= [-te-st/s]0∞ + (1/s)∫0∞ e-st dt = 0 + (1/s)(1/s) = 1/s².
More generally: L{tn} = n!/sn+1.
Example 4: L{sin(at)} and L{cos(at)}
Using integration by parts twice (or Euler's formula):
L{sin(at)} = a/(s² + a²), s > 0.
L{cos(at)} = s/(s² + a²), s > 0.
Linearity
Linearity Property: L{αf(t) + βg(t)} = αF(s) + βG(s). The Laplace transform of a linear combination is the same linear combination of the transforms.
Example 5: Using Linearity
Find L{3t² - 5e2t + 7sin(4t)}.
Step 1: L{t²} = 2!/s³ = 2/s³.
Step 2: L{e2t} = 1/(s-2).
Step 3: L{sin 4t} = 4/(s²+16).
Answer: L{3t² - 5e2t + 7sin 4t} = 6/s³ - 5/(s-2) + 28/(s²+16).
Table of Basic Laplace Transforms
| f(t) | F(s) = L{f(t)} | Condition |
|---|---|---|
| 1 | 1/s | s > 0 |
| tn | n!/sn+1 | s > 0 |
| eat | 1/(s-a) | s > a |
| sin(at) | a/(s²+a²) | s > 0 |
| cos(at) | s/(s²+a²) | s > 0 |
| eatsin(bt) | b/[(s-a)²+b²] | s > a |
| eatcos(bt) | (s-a)/[(s-a)²+b²] | s > a |
| tneat | n!/(s-a)n+1 | s > a |
Existence of the Transform
Existence Theorem: If f(t) is piecewise continuous on [0, ∞) and of exponential order (|f(t)| ≤ Mect for some constants M, c), then L{f(t)} exists for s > c.
All polynomials, exponentials, sines, cosines, and their combinations satisfy these conditions. Functions like et² do not (they grow too fast).
Check Your Understanding
1. Compute L{5 + 2t - e-3t}.
2. What is L{t³}?
3. Find L{e2tcos(3t)}.
4. Does L{et²} exist? Why or why not?
Key Takeaways
- The Laplace transform L{f(t)} = ∫0∞ e-stf(t) dt converts functions of t to functions of s.
- Key transforms: L{1}=1/s, L{tn}=n!/sn+1, L{eat}=1/(s-a).
- Trig: L{sin at}=a/(s²+a²), L{cos at}=s/(s²+a²).
- Linearity lets you transform term by term.
- The transform exists for functions of exponential order.