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:<math>\lim_{b\to\infty} e^{-sb}f(b)=0</math>
:<math>\lim_{b\to\infty} e^{-sb}f(b)=0</math>
==Examples==
==Examples==
===Example 1===
===Example 1===
:<math>\int_0^{\infty}e^{-st+at} \,dt=\frac{1}{a-s}</math>
:<math>\int_0^{\infty}e^{-st+at} \,dt=\frac{1}{a-s}</math>
-->
Substituting the inverse transform we have
:<math>\int_0^{\infty}e^{-st}y\,dt=\frac{1}{a+1}\left(\int_0^{\infty}e^{-st}e^{at}\,dt-\int_0^{\infty}e^{-st}e^{-t}\,dt \right) </math>
:<math>\int_0^{\infty}e^{-st}y\,dt=\int_0^{\infty}e^{-st}\left[\frac{1}{a+1}\left(e^{at}-e^{-t} \right)\right]\,dt </math>
Which leads to the answer
:<math>y=\frac{1}{a+1}\left(e^{at}-e^{-t} \right)</math>
==References==
==References==
*D. Lomen and D. Lovelock, ''Differential Equations Graphics. Model. Data.'', John Wiley and Sons, Toronto, 1999.
*D. Lomen and D. Lovelock, ''Differential Equations Graphics. Model. Data.'', John Wiley and Sons, Toronto, 1999.
*[http://mathworld.wolfram.com/LaplaceTransform.html Laplace transform] on Wolfram Mathworld
*[http://mathworld.wolfram.com/LaplaceTransform.html Laplace transform] on Wolfram Mathworld