Laplace Transform Reference Cheat Sheet

Key Facts

• The Laplace transform is defined by $\mathcal{L}\{f(t)\}=F(s)=\int_{0}^{\infty} e^{-st}f(t)\,dt$ when the integral converges.
• Linearity means $\mathcal{L}\{af(t)+bg(t)\}=aF(s)+bG(s)$ for constants $a$ and $b$.
• The basic power rule is $\mathcal{L}\{t^n\}=\frac{n!}{s^{n+1}}$ for integers $n\ge 0$.
• The exponential shift rule is $\mathcal{L}\{e^{at}f(t)\}=F(s-a)$.
• Derivative transforms are $\mathcal{L}\{f'(t)\}=sF(s)-f(0)$ and $\mathcal{L}\{f''(t)\}=s^2F(s)-sf(0)-f'(0)$.
• The first shifting theorem for step functions is $\mathcal{L}\{u(t-a)f(t-a)\}=e^{-as}F(s)$ for $a>0$.
• Convolution satisfies $\mathcal{L}\{(f*g)(t)\}=F(s)G(s)$, where $(f*g)(t)=\int_0^t f(\tau)g(t-\tau)\,d\tau$.
• Common transforms include $\mathcal{L}\{1\}=\frac{1}{s}$, $\mathcal{L}\{e^{at}\}=\frac{1}{s-a}$, $\mathcal{L}\{\sin bt\}=\frac{b}{s^2+b^2}$, and $\mathcal{L}\{\cos bt\}=\frac{s}{s^2+b^2}$.

Vocabulary

Laplace transform

An integral transform that changes a time-domain function $f(t)$ into a frequency-domain function $F(s)$.

Inverse Laplace transform

The operation $\mathcal{L}^{-1}\{F(s)\}=f(t)$ that recovers the original time-domain function from its transform.

Unit step function

The function $u(t-a)$ equals $0$ for $t<a$ and $1$ for $t\ge a$, so it models a delayed switch.

Convolution

The convolution $(f*g)(t)=\int_0^t f(\tau)g(t-\tau)\,d\tau$ combines two functions in a way that becomes multiplication after transforming.

Initial value problem

A differential equation together with starting values such as $y(0)$ and $y'(0)$.

Region of convergence

The set of $s$-values for which the improper integral defining $\mathcal{L}\{f(t)\}$ converges.