Skip to main content
Mathematics LibreTexts

Elementary Laplace Transforms

  • Page ID
    618
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \mathcal{L} \{1\}(s)\) = \( \dfrac{1}{s} \; (s>0)\) (1)
    \( \mathcal{L} \{e^{at}\}(s)\) = \( \dfrac{1}{s-a} \; (s>a)\) (2)
    \( \mathcal{L} \{t^{n}\}(s)\) = \( \dfrac{n!}{s^{n+1}} \;\; (s>0, n \text{ is a positive integer})\) (3)
    \( \mathcal{L} \{t^{p}\}(s)\) = 4)
    \( \mathcal{L} \{\sin (at)\}(s)\) = \( \dfrac{a}{s^2+a^2} \; (s>0)\) (5)
    \( \mathcal{L} \{\cos (at)\}(s)\) = \( \dfrac{s}{s^2+a^2} \; (s>0)\) (6)
    \( \mathcal{L} \{e^{at} \cdot \sin (bt)\}(s)\) = \( \dfrac{b}{(s-a)^2+b^2} \; (s>a)\) (7)
    \( \mathcal{L} \{e^{at} \cdot \cos (bt)\}(s)\) = \( \dfrac{s-a}{(s-a)^2+b^2} \; (s>a)\) (8)
    \( \mathcal{L} \{t^{n} \cdot e^{at}\}(s)\) = \( \dfrac{n!}{(s-a)^{n+1}} \; (s>a)\) (9)
    \( \mathcal{L} \{t^{n} \cdot f(t)\}(s)\) = \( (-1)^n \dfrac{d^n}{ds^n} \mathcal{L} \{f(t)\}(s)\) (10)
    \( \mathcal{L} \{f'(t)\}(s)\) = \( s \cdot \mathcal{L} \{f(t) \} - f(0)\) (11)
    \( \mathcal{L} \{H_c(t)\}(s)\) = \( \dfrac{e^{-cs}}{s} \; (s>0)\) (12)
    \( \mathcal{L} \{H_c(t) \cdot f(t-c)\}(s)\) = \( e^{-cs} \mathcal{L}\{f(t)\}(s)\) (13)
    \( \mathcal{L} \{H_c(t) \cdot f(t)\}(s)\) = \( e^{-cs} \mathcal{L}\{f(t+c)\}(s)\) (14)
    \( \mathcal{L} \{ \delta_c(t)\}(s)\) = \( e^{-cs}\) (15)
    \( \mathcal{L} \{e({ct} cdot f(t)\}(s)\) = \( \mathcal{L} \{f(t)\}(s-c)\) (16)



    Elementary Laplace Transforms is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

    • Was this article helpful?