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Engineering LibreTexts

11.2: Common Laplace Transforms

  • Page ID
    22908
  • Table \(\PageIndex{1}\)
    Signal Laplace Transform Region of Convergence
    \(\delta(t)\) \(1\) All \(s\)
    \(\delta(t-T)\) \(e^{(-sT)}\) All \(s\)
    \(u(t)\) \(\frac{1}{s}\) \(\operatorname{Re}(s)>0\)
    \(u(-t)\) \(\frac{1}{s}\) \(\operatorname{Re}(s)<0\)
    \(tu(t)\) \(\frac{1}{s^2}\) \(\operatorname{Re}(s)>0\)
    \(t^{n} u(t)\) \(\frac{n !}{s^{n+1}}\) \(\operatorname{Re}(s)>0\)
    \(-(t^n u(-t))\) \(\frac{n !}{s^{n+1}}\) \(\operatorname{Re}(s)<0\)
    \(e^{-(\lambda t)} u(t)\) \(\frac{1}{s+\lambda}\) \(\operatorname{Re}(s)>-\lambda\)
    \(\left(-e^{-(\lambda t)}\right) u(-t)\) \(\frac{1}{s+\lambda}\) \(\operatorname{Re}(s)<-\lambda\)
    \(t e^{-(\lambda t)} u(t)\) \(\frac{1}{(s+\lambda)^{2}}\) \(\operatorname{Re}(s)>-\lambda\)
    \(t^{n} e^{-(\lambda t)} u(t)\) \(\frac{n !}{(s+\lambda)^{n+1}}\) \(\operatorname{Re}(s)>-\lambda\)
    \(-\left(t^{n} e^{-(\lambda t)} u(-t)\right)\) \(\frac{n !}{(s+\lambda)^{n+1}}\) \(\operatorname{Re}(s)<-\lambda\)
    \(\cos (b t) u(t)\) \(\frac{s}{s^{2}+b^{2}}\) \(\operatorname{Re}(s)>0\)
    \(\sin (b t) u(t)\) \(\frac{b}{s^{2}+b^{2}}\) \(\operatorname{Re}(s)>0\)
    \(e^{-(a t)} \cos (b t) u(t)\) \(\frac{s+a}{(s+a)^{2}+b^{2}}\) \(\operatorname{Re}(s)>-a\)
    \(e^{-(a t)} \sin (b t) u(t)\) \(\frac{b}{(s+a)^{2}+b^{2}}\) \(\operatorname{Re}(s)>-a\)
    \(\frac{d^{n}}{d t^{n}} \delta(t)\) \(s^n\) All \(s\)