Table of Laplace Transforms
This is a table of bare functions. Using these functions in addition to the results in other tables, we can build up more complicated functions
| Given Function | Laplace Transform | Condition |
|---|---|---|
| $0$ | $0$ | |
| $1$ | $\frac{1}{s}$ | $\lvert s \rvert > 0$ |
| $t^n$ | $\frac{n!}{s^n}$ | $\lvert s \rvert > 0, \quad n \in \mathbb{N}$ |
| $t^{-n}$ | divergent | $n \in \mathbb{N}$ |
| $e^t$ | $\frac{1}{s-1}$ | $s \neq 1$ |
| $\sin t$ | $\frac{1}{s^2+1}$ | $s \neq i$ |
| $\cos t$ | $\frac{s}{s^2+1}$ | $s \neq i$ |
| $\sinh t$ | $\frac{1}{s^2-1}$ | $s \neq 1$ |
| $\cosh t$ | $\frac{s}{s^2-1}$ | $s \neq 1$ |
Now I give a table that lets us manipulate Laplace transformation addition and scalar multiplication
| Given Function | Laplace Transform | Condition |
|---|---|---|
| $f(t) \pm g(t)$ | $F(s) \pm G(s)$ | |
| $\alpha f(t)$ | $\alpha F(s)$ | |
| $f(-t)$ | $F(-s)$ | |
| $f(bt)$ | $\frac{1}{\lvert b \rvert}F(s/b)$ | $b \neq 0$ |
| $f(t-c)$ | $e^{-cs}F(s)$ | $f(t) = 0 \quad \forall t < 0$ |
| $e^{\alpha t} f(t)$ | $F(s-\alpha)$ | $\lvert s \rvert > \lvert \alpha \rvert$ |
Now, I give function multiplication. The trig ones aren’t super useful, but I prove them in the series.
| Given Function | Laplace Transform | Condition |
|---|---|---|
| $\sin(t) f(t)$ | $\frac{1}{2i}(F(s+i) - F(s-i))$ | |
| $\cos(t) f(t)$ | $\frac{1}{2}(F(s+i) + F(s-i))$ | |
| $\sinh(t) f(t)$ | $\frac{1}{2}(F(s+1) - F(s-1))$ | |
| $\cosh(t) f(t)$ | $\frac{1}{2}(F(s+1) + F(s-1))$ | |
| $\theta(t-c)f(t-c)$ | $e^{-cs} F(s)$ | |
| $\delta(t-c)f(t-d)$ | $e^{-cs} f(c-d)$ | $c \geq 0$ |
Next, I provide the interaction between Laplace transforms, derivatives, and integrals
| Given Function | Laplace Transform | Condition |
|---|---|---|
| $t^n f(t)$ | $(-1)^n \frac{d^n}{d^n s} F(s)$ | $n \in \mathbb{N}$ $F$ is $n$-times differentiable $f$ is continuous |
| $t^{-n} f(t)$ | $\int_{s}^{\infty} \cdots \int_{s}^{\infty} F(u) \ du_1 \cdots \ du_n$ | $n \in \mathbb{N}$ $F$ is $n$-times integrable $f$ is continuous |
| $\frac{d^n}{d^n t} f(t)$ | $s^n F(s) - \sum_{k=0}^{n-1} s^{n-k-1} f^{(k)}(0)$ | $n \in \mathbb{N}$ $f$ is $n$-times differentiable $\frac{d^k}{d^k t} f(t) = o(e^t) \quad \forall k < n$ |
| $\int_0^s \cdots \int_0^s f(t) \ du_1 \cdots \ du_n$ | $F(s) / s^n$ | $n \in \mathbb{N}$ $F$ is $n$-times integrable $\int_0^s \cdots \int_0^s f(t) \ du_1 \cdots \ du_n = o(e^t) \quad \forall k < n$ |
Finally, we have two miscellaneous identities for periodic functions and convolution
| Given Function | Laplace Transform | Condition |
|---|---|---|
| $f(t)$ | $\frac{1}{1 - e^{-Ts}} \int_{0}^T f(t) e^{-st} \ dt$ | $f(t)$ is $T$-periodic |
| $f(t) * g(t)$ | $F(s) \cdot G(s)$ | $f(t) = g(t) = 0 \quad \forall t < 0$ |