Graduationwoot

Dragon Notes

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  UNDER CONSTRUCTION

Laplace Transform Pairs


Eq\(x(t)\)\(\bn{X}(\bn{s})=\mathcal{L} [x(t)]\)
\([1]\)\(\ds \delta \left(t\right)\)\(\ds 1\)
\([2]\)\(\ds u\left(t\right)\)\(\ds \frac{1}{\bn{s}}\)
\([3]\)\(\ds {\mathrm{sin} \left({\omega }_0t\right)\ }u\left(t\right)\)\(\ds \frac{{\omega }_0}{{\bn{s}}^{\bn{2}}+{\omega }^2_0}\)
\([4]\)\(\ds {\mathrm{cos} \left({\omega }_0t\right)\ }u\left(t\right)\)\(\ds \frac{\bn{s}}{{\bn{s}}^{\bn{2}}\mathrm{+}{\mathrm{\omega }}^2_0}\)
\([5]\)\(\ds {\mathrm{sin} \left({\omega }_0t+\theta \right)\ }u\left(t\right)\)\(\ds \frac{\bn{s}\ {\mathrm{sin} \left(\theta \right)\ }+{\omega }_0{\mathrm{cos} \left(\theta \right)\ }}{{\bn{s}}^{\bn{2}}\mathrm{+}{\mathrm{\omega }}^2_0}\)
\([6]\)\(\ds {\mathrm{cos} \left({\omega }_0t+\theta \right)\ }u\left(t\right)\)\(\ds \frac{\bn{s}\ {\mathrm{cos} \left(\theta \right)\ }-{\omega }_0{\mathrm{sin} \left(\theta \right)\ }}{{\bn{s}}^{\bn{2}}\mathrm{+}{\mathrm{\omega }}^2_0}\)
\([7]\)\(\ds \delta (t-T)\)\(\ds e^{-T\bn{s}}\)
\([8]\)\(\ds u\left(t-T\right)\)\(\ds \frac{e^{-T\bn{s}}}{\bn{s}}\)
\([9]\)\(\ds e^{-at}u\left(t\right)\)\(\ds \frac{1}{\bn{s}\mathrm{+}a}\)
\([10]\)\(\ds e^{-a\left(t-T\right)}u\left(t-T\right)\)\(\ds \frac{e^{-T\bn{s}}}{\bn{s}+a}\)
\([11]\)\(\ds tu\left(t\right)\)\(\ds \frac{1}{{\bn{s}}^{\bn{2}}}\)
\([12]\)\(\ds \left(t-T\right)u\left(t-T\right)\)\(\ds \frac{e^{-T\bn{s}}}{{\bn{s}}^{\bn{2}}}\)
\([13]\)\(\ds t^2u\left(t\right)\)\(\ds \frac{2}{{\bn{s}}^{\bn{3}}}\)
\([14]\)\(\ds te^{-at}u\left(t\right)\)\(\ds \frac{1}{{\left(\bn{s}+a\right)}^2}\)
\([15]\)\(\ds t^2e^{-at}u\left(t\right)\)\(\ds \frac{2}{{\left(\bn{s}+a\right)}^3}\)
\([16]\)\(\ds t^{n-1}e^{-at}u\left(t\right)\)\(\ds \frac{\left(n-1\right)!}{{\left(\bn{s}+a\right)}^n}\)
\([17]\)\(\ds e^{-at}{\mathrm{sin} \left({\omega }_0t\right)\ }u\left(t\right)\)\(\ds \frac{{\omega }_0}{{(\bn{s}\bn{+}a)}^{\bn{2}}+{\omega }^2_0}\)
\([18]\)\(\ds e^{-at}{\mathrm{cos} \left({\omega }_0t\right)\ }u\left(t\right)\)\(\ds \frac{\left(\bn{s}\mathrm{+}a\right)}{{\left(\bn{s}\mathrm{+}a\right)}^{\bn{2}}\mathrm{+}{\mathrm{\omega }}^2_0}\)
\([19]\)\(\ds 2e^{-at}{\mathrm{cos} \left(bt-\theta \right)\ }u\left(t\right)\)\(\ds \frac{e^{j\theta }}{\bn{s}+a+jb}+\frac{e^{-j\theta }}{\bn{s}+a-jb}\)
\([20]\)\(\ds e^{-at}{\mathrm{cos} \left(bt-\theta \right)\ }u\left(t\right)\)\(\ds \frac{\left(\bn{s}\mathrm{+}a\right){\mathrm{cos} \left(\theta \right)}+b\ {\mathrm{sin} \left(\theta \right)\ }}{\left(\bn{s}\mathrm{+}a\right)^{\mathrm{2}}\mathrm{+}b^2}\)
\([21]\)\(\ds \frac{2t^{n-1}}{\left(n-1\right)!}e^{-at}{\mathrm{cos} \left(bt-\theta \right)\ }u\left(t\right)\)\(\ds \frac{e^{j\theta }}{{\left(\bn{s}+a+jb\right)}^n}+\frac{e^{-j\theta }}{{\left(\bn{s}+a-jb\right)}^n}\)


Dragon Notes,   Est. 2018     About

By OverLordGoldDragon