Laplace Transform Determination/Series Method

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Solution Technique for Laplace Transform

To find the Laplace transform of a function $f$, one can evaluate it as follows:


Let $\map f t$ have a power series expansion given by:

\(\ds \map f t\) \(=\) \(\ds a_0 + a_1 t + a_2 t^2 + \dotsb\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty a_n t^n\)


Then the Laplace transform of $f$ can be found by taking the sum of the Laplace transforms of each term in the series:

\(\ds \laptrans {\map f t}\) \(=\) \(\ds \dfrac {a_0} s + \dfrac {a_1} {s^2} + \dfrac {a_2} {s^3} + \dotsb\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \dfrac {n! \, a_n} {s^{n + 1} }\)

if $\laptrans {\map f t}$ is convergent for $s > \gamma$.


Proof

From Laplace Transform of Positive Integer Power:

$\laptrans {t^n} = \dfrac {n!} {s^{n + 1} }$


Then it is seen that $\ds \laptrans {\sum_{n \mathop = 0}^\infty a_n t^n}$ is a Linear Combination of Laplace Transforms.

The result follows.

$\blacksquare$


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Examples

Laplace Transform Determination/Series Method/Examples

Sources

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