Most Laplace tables I have found online have either too little information or do not meet my standards for typesetting, so I made my own.

## Source Code#

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57  \documentclass[12pt]{article} \usepackage[margin=0.25in]{geometry} \usepackage{tabularx, amsmath, mathrsfs, physics} \begin{document} \thispagestyle{empty} \large{\begin{center}\textbf{Table of Laplace Transforms}\end{center}} \normalsize \begin{equation*} F(s)=\mathscr{L}\left\{f(t)\right\}=\int_{0}^{\infty}f(t)e^{-st}\dd{t} \end{equation*} \begin{center} \begin{tabularx}{\linewidth}{rXc|rXc} & $f(t)=\mathscr{L}^{-1}\left\{F(s)\right\}$ & $F(s)=\mathscr{L}\left\{f(t)\right\}$ & & $f(t)=\mathscr{L}^{-1}\left\{F(s)\right\}$ & $F(s)=\mathscr{L}\left\{f(t)\right\}$ \\ \hline \rule{0pt}{5.5ex} 1. & $1$ & $\dfrac{1}{s}$ & 2. & $e^{at}$ & $\dfrac{1}{s-a}$ \\ \rule{0pt}{5.5ex} 3. & $t^n,\;n=1,2,3,\dotsc$ & $\dfrac{n!}{s^{n+1}}$ & 4. & $t^p,\;p>-1$ & $\dfrac{\Gamma(p+1)}{s^{p+1}}$ \\ \rule{0pt}{5.5ex} 5. & $\sqrt{t}$ & $\dfrac{\sqrt{\pi}}{2s^\frac{3}{2}}$ & 6. & $t^{n-\frac{1}{2}},\;n=1,2,3,\dotsc$ & $\dfrac{1\cdot3\cdot5\dotsm(2n-1)\sqrt{\pi}}{2^ns^{n+\frac{1}{2}}}$ \\ \rule{0pt}{5.5ex} 7. & $\sin(at)$ & $\dfrac{a}{s^2+a^2}$ & 8. & $\cos(at)$ & $\dfrac{s}{s^2+a^2}$ \\ \rule{0pt}{5.5ex} 9. & $t\sin(at)$ & $\dfrac{2as}{\left(s^2+a^2\right)^2}$ & 10. & $t\cos(at)$ & $\dfrac{s^2-a^2}{\left(s^2+a^2\right)^2}$ \\ \rule{0pt}{5.5ex} 11. & $\sin(at)-at\cos(at)$ & $\dfrac{2a^3}{\left(s^2+a^2\right)^2}$ & 12. & $\sin(at)+at\cos(at)$ & $\dfrac{2as^2}{\left(s^2+a^2\right)^2}$ \\ \rule{0pt}{5.5ex} 13. & $\cos(at)-at\sin(at)$ & $\dfrac{s\left(s^2-a^2\right)}{\left(s^2+a^2\right)^2}$ & 14. & $\cos(at)+at\sin(at)$ & $\dfrac{s\left(s^2+3a^2\right)}{\left(s^2+a^2\right)^2}$ \\ \rule{0pt}{5.5ex} 15. & $\sin(at+b)$ & $\dfrac{s\sin(b)+a\cos(b)}{s^2+a^2}$ & 16. & $\cos(at+b)$ & $\dfrac{s\cos(b)-a\sin(b)}{s^2+a^2}$ \\ \rule{0pt}{5.5ex} 17. & $\sinh(at)$ & $\dfrac{a}{s^2-a^2}$ & 18. & $\cosh(at)$ & $\dfrac{s}{s^2-a^2}$ \\ \rule{0pt}{5.5ex} 19. & $e^{at}\sin(bt)$ & $\dfrac{b}{(s-a)^2+b^2}$ & 20. & $e^{at}\cos(bt)$ & $\dfrac{s-a}{(s-a)^2+b^2}$ \\ \rule{0pt}{5.5ex} 21. & $e^{at}\sinh(bt)$ & $\dfrac{b}{(s-a)^2-b^2}$ & 22. & $e^{at}\cosh(bt)$ & $\dfrac{s-a}{(s-a)^2-b^2}$ \\ \rule{0pt}{5.5ex} 23. & $t^ne^{at}\;,n=1,2,3,\dotsc$ & $\dfrac{n!}{(s-a)^{n+1}}$ & 24. & $f(ct)$ & $\dfrac{1}{c}F\left(\dfrac{s}{c}\right)$ \\ \rule{0pt}{5.5ex} 25. & $u_c(t)=u(t-c)$ & $\dfrac{e^{-cs}}{s}$ & 26. & $\delta(t-c)$ & $e^{-cs}$ \\ \rule{0pt}{5.5ex} 27. & $u_c(t)f(t-c)$ & $e^{-cs}F(s)$ & 28. & $u_c(t)g(t)$ & $e^{-cs}\mathscr{L}\left\{g(t+c)\right\}$ \\ \rule{0pt}{5.5ex} 29. & $e^{ct}f(t)$ & $F(s-c)$ & 30. & $t^nf(t),\;n=1,2,3,\dotsc$ & $(-1)^nF^{(n)}(s)$ \\ \rule{0pt}{5.5ex} 31. & $\dfrac{1}{t}f(t)$ & $\int_{s}^{\infty}F(u)\dd{u}$ & 32. & $\int_{0}^{t}f(v)\dd{v}$ & $\dfrac{F(s)}{s}$ \\ \rule{0pt}{5.5ex} 33. & $\int_{0}^{t}f(t-\tau)g(\tau)\dd{\tau}$ & $F(s)G(s)$ & 34. & $f(t+T)=f(t)$ & $\dfrac{\int_{0}^{T}e^{-st}f(t)\dd{t}}{1-e^{-sT}}$ \\ \rule{0pt}{5.5ex} 35. & $\dot{f}(t)$ & $sF(s)-f(0^+)$ & 36. & $\ddot{f}(t)$ & $s^2F(s)-sf(0^+)-\dot{f}(0^+)$ \\ \end{tabularx} \end{center} \end{document}