# Laplace transform of $$\displaystyle{f{{\left({t}^{{2}}\right)}}}$$ Suppose we know the

Quinn Moses 2022-04-16 Answered
Laplace transform of $f\left({t}^{2}\right)$
Suppose we know the Laplace transform of a function f(t):
$F\left(s\right)={\int }_{0}^{\mathrm{\infty }}f\left(t\right){e}^{-st}dt$
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muthe2ulj
You can change variables ${t}^{2}=x$
$H\left(s\right)={\int }_{0}^{\mathrm{\infty }}f\left({t}^{2}\right){e}^{-st}dt={\int }_{0}^{\mathrm{\infty }}\frac{f\left(x\right)}{2\sqrt{x}}{e}^{-s\sqrt{x}}dx$
Now we can express $\frac{{e}^{-s\sqrt{x}}}{\sqrt{x}}$ as a Laplace transform, so:

$={\int }_{0}^{\mathrm{\infty }}\frac{F\left(t\right)}{2\sqrt{\pi t}}{e}^{-\frac{{s}^{2}}{4t}}dt$