# Thread: laplace transform of t^(3/2)

1. ## laplace transform of t^(3/2)

i know what the laplace transform of t^(3/2) is from looking at a table but how do you prove it? i applied the definition of the laplace transform and ended up doing integration by parts 2 times so that the integral i ended up with has e^(-st)*t^(-1/2). when i expanded again using integration by parts, the (-1/s)e^(-st)*t^(-1/2) evaluates to negative infinite which means it diverges. so i can't proceed further. how do you derive the laplace transform of t^(3/2)?

2. So we have

$L\{t^{3/2}\}=\int_{0}^{\infty}t^{3/2}e^{-st}dt.$

Make the substitution $u=st$ to get

$L\{t^{3/2}\}=\int_{0}^{\infty}\left(\frac{u}{s}\right)^{3/2}e^{-u}\frac{du}{s}=\frac{1}{s^{5/2}}\int_{0}^{\infty}u^{3/2}e^{-u}du$.

The integral is now in the form of a gamma function

$\frac{1}{s^{5/2}}\int_{0}^{\infty}u^{3/2}e^{-u}du=\frac{1}{s^{5/2}}\Gamma\left(\frac{5}{2}\right)=\frac{1}{s^{5/2}}\cdot \frac{3}{2} \cdot \frac{1}{2} \cdot \Gamma\left(\frac{1}{2}\right)=\frac{3\sqrt{\pi}}{ 4s^{5/2}}$.

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# what is the laplace of t^(3/2)

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