# Compute the following Laplace transforms. t^{\frac{3}{2}}

Compute the following Laplace transforms
$$\displaystyle{t}^{{{\frac{{{3}}}{{{2}}}}}}$$

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To find laplace transform of
$$\displaystyle{f{{\left({t}\right)}}}={t}^{{{\frac{{{3}}}{{{2}}}}}}$$
Here we use
$$\displaystyle{L}{\left[{t}^{{\alpha}}\right]}_{{s}}={\frac{{\Gamma{\left(\alpha+{1}\right)}}}{{{s}^{{\alpha+{1}}}}}},-{1}{<}\alpha$$
$$\displaystyle\Gamma{\left(\alpha+{1}\right)}=\alpha\Gamma{\left(\alpha\right)}$$
$$\displaystyle\Gamma{\left({\frac{{{1}}}{{{2}}}}\right)}=\sqrt{{\pi}}$$
$$\displaystyle\therefore{L}{\left[{t}^{{{\frac{{{3}}}{{{2}}}}}}\right]}={\frac{{\Gamma{\left({\frac{{{3}}}{{{2}}}}+{1}\right)}}}{{{s}^{{{\frac{{{3}}}{{{2}}}}+{1}}}}}}$$
$$\displaystyle={\frac{{{\frac{{{3}}}{{{2}}}}\Gamma{\left({\frac{{{3}}}{{{2}}}}\right)}}}{{{s}^{{{\frac{{{5}}}{{{2}}}}}}}}}$$
$$\displaystyle={\frac{{{\frac{{{3}}}{{{2}}}}\Gamma{\left({\frac{{{1}}}{{{2}}}}+{1}\right)}}}{{{s}^{{{\frac{{{5}}}{{{2}}}}}}}}}$$
$$\displaystyle={\frac{{{\frac{{{3}}}{{{2}}}}\times{\frac{{{1}}}{{{2}}}}\Gamma{\left({\frac{{{1}}}{{{2}}}}\right)}}}{{{s}^{{{\frac{{{5}}}{{{2}}}}}}}}}$$
$$\displaystyle={\frac{{{\frac{{{3}}}{{{4}}}}\cdot\sqrt{{\pi}}}}{{{s}^{{{\frac{{{5}}}{{{2}}}}}}}}}={\frac{{{3}\sqrt{{\pi}}}}{{{4}{s}^{{{\frac{{{5}}}{{{2}}}}}}}}}$$
$$\displaystyle\therefore{L}{\left[{t}^{{{\frac{{{3}}}{{{2}}}}}}\right]}={\frac{{{3}\sqrt{{\pi}}}}{{{4}{s}^{{{\frac{{{5}}}{{{2}}}}}}}}}$$