Suppose we know the Laplace transform of a function f(t):

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):

$F\left(s\right)={\int}_{0}^{\mathrm{\infty}}f\left(t\right){e}^{-st}dt$

Suppose we know the Laplace transform of a function f(t):

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muthe2ulj

Answered 2022-04-17
Author has **10** answers

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:

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

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

Now we can express

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