#+title:Laplace Transformations
#+author: Thomas Albers Raviola
#+date: 2022-10-01
#+setupfile: ../../math_options.org

\begin{align*}
\delta(t) & 1\\
\delta(t - a) & e^{as}\\
1 & \frac{1}{s}\\
t & \frac{1}{s^2}\\
\frac{t^{n-1}}{(n-1)!} & \frac{1}{s^n}, n \in \mathbb{N}\\
\frac{t^{a-1}}{\Gamma(a)} & \frac{1}{s^a}\\
e^{-at} &  \frac{1}{s+a}\\
\frac{t^{n-1}e^{-at}}{(n-1)!} & \frac{1}{(s+a)^n}, n \in \mathbb{N}\\
\frac{e^{-at}-e^{-bt}}{b - a} & \frac{1}{(s+a)(s+b)}, a \neq b\\
\frac{1}{a}\sin(at) & \frac{1}{s^2 + a^2}\\
\cos(at) & \frac{s}{s^2+a^2}\\
\frac{1}{a}\sinh(at) & \frac{1}{s^2-a^2}\\
\cosh(at) & \frac{s}{s^2-a^2}\\
\frac{1-\cos(at)}{a^2} & \frac{1}{s(s^2+a^2)}\\
\frac{at - \sin(at)}{a^3} & \frac{1}{s^2(s^2 + a^2)}\\
\frac{\sin(at) - at\cos(at)}{2a^3} & \frac{1}{(s^2 + a^2)^2}\\

\frac{t\sin(at)}{2a} & \frac{s}{(s^2+a^2)^2}\\
\frac{\sin(at) + at\cos(at)}{2a} & \frac{s^2}{(s^2 + a^2)^2}\\
\frac{b\sin(at) - a\sin(bt)}{ab(b^2 - a^2)} & \frac{1}{(s^2+a^2)(s^2+b^2)}, a^2 \neq b^2
\frac{\cos(at) - \cos(bt)}{b^2 - a^2} & \frac{s}{(s^2+a^2)(s^2+b^2)}, a^2 \neq b^2
\frac{1}{b}e^{-at}\sin(bt) & \frac{1}{(s+a)^2 + b^2}\\
e^{-at}\cos(bt) & \frac{s+a}{(s+a)^2 + b^2}\\
\frac{\sinh(at) - \sin(at)}{2a^3} & \frac{1}{s^4 - a^4}\\
\frac{\sin(at)\sinh(at)}{2a^2} & \frac{s}{s^4+4a^4}\\
\frac{1}{\sqrt{\pi t}} & \frac{1}{\sqrt{s}}\\
\frac{\sin{at}{t}} & \arctan{\frac{a}{s}}\\
u(t) - u(t-k) & \frac{1-e^{-ks}}{s}\\
\frac{(t-k)^{a-1}}{\Gamma(a)}u(t-k) & \frac{1}{s^a}e^{-ks}, a > 0\\
\sum_{n=0}^\inf u(t-nk) & \frac{1}{s(1-e^{-ks})}\\
\frac{1}{2}(\sin(t) + \|\sin(t)\|) & \frac{1}{(s^2 + 1)(1 - e^{\pi s})}\\
\|\sin(at)\| & \frac{a\coth(\frac{\pi s}{2 a})}{s^2 + a^2}
\end{align*}