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

boitshupoO 2021-09-14 Answered
Compute the following Laplace transforms
\(\displaystyle{t}^{{{\frac{{{3}}}{{{2}}}}}}\)

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Expert Answer

Khribechy
Answered 2021-09-15 Author has 13406 answers

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

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