The Laplace Transform
An improper integral converts a function of time into a function of a new variable, turning differentiation into multiplication. Its usefulness rests on the transform being reversible.
What a learner can do afterwards
- Compute the transform of a simple function from the defining integral
- Use the derivative rule to see why the transform turns calculus into algebra
- Read an inverse transform off a table after partial fractions
1 · Read
You start with a function of time, f(t). Multiply it by e to the power minus s t. Then integrate that product from zero to infinity. The result is a new function of s, written F(s). This recipe is the Laplace transform.
Take f(t) equal to e to the power a t. The integral becomes e to the power (a minus s) t from zero to infinity. That works out to 1 over (s minus a). It converges only when s is bigger than a. For e to the power 3t, the transform is 1 over (s minus 3), valid for s above 3.
Differentiation in the t world becomes multiplication in the s world. The rule is that L of f prime equals s times F(s) minus f(0). Because of it, a differential equation turns into an algebra equation. You solve for F(s) with ordinary algebra, using the starting values directly.
To go back, split F(s) into simple pieces with partial fractions. Then match each piece to a table entry. For example, 1 over ((s minus 1)(s plus 2)) splits into one third over (s minus 1) minus one third over (s plus 2). Reading the table backward gives one third e to the t minus one third e to the minus 2t.
The Laplace transform trades a time function for an s function, swaps derivatives for multiplication, and inverts through partial fractions and a table.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.