Linear Combination of Laplace Transforms

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Theorem

Let $\laptrans f$ denote the Laplace transform of the real function $f$.

Let $f, g$ be functions such that $\laptrans f$ and $\laptrans g$ exist.

Let $\lambda, \mu \in \C$ or $\R$ be constants.


Then:

$\laptrans {\lambda \, \map f t + \mu \, \map g t} = \lambda \laptrans {\map f t} + \mu \laptrans {\map g t}$

everywhere all the above expressions are defined.


General Result

Linear Combination of Laplace Transforms/General Result

Proof

\(\ds \laptrans {\lambda \, \map f t + \mu \, \map g t}\) \(=\) \(\ds \int_0^{\to +\infty} e^{-s t} \paren {\lambda \, \map f t + \mu \, \map g t} \rd t\) Definition of Laplace Transform
\(\ds \) \(=\) \(\ds \lim_{A \mathop \to +\infty} \paren {\int_0^A e^{-s t} \paren {\lambda \, \map f t + \mu \, \map g t} \rd t}\) Definition of Improper Integral
\(\ds \) \(=\) \(\ds \lim_{A \mathop \to +\infty} \paren {\lambda \int_0^A e^{-s t} \map f t \rd t + \mu \int_0^A e^{-s t} \map g t \rd t}\) distributing $e^{-s t}$, Linear Combination of Complex Integrals
\(\ds \) \(=\) \(\ds \lambda \lim_{A \mathop \to +\infty} \int_0^A e^{-s t} \map f t \rd t + \mu \lim_{A \mathop \to +\infty} \int_0^A e^{-s t} \map g t \rd t\) Combination Theorem for Limits at Infinity
\(\ds \) \(=\) \(\ds \lambda \int_0^{\to +\infty} e^{-s t} \map f t \rd t + \mu \int_0^{\to +\infty} e^{-st} \map g t \rd t\) Definition of Improper Integral
\(\ds \) \(=\) \(\ds \lambda \laptrans {\map f t} + \mu \laptrans {\map g t}\) Definition of Laplace Transform

$\blacksquare$


Examples

Example $1$

$\laptrans {4 t^2 - 3 \cos 2 t + 5 e^{-t} } = \dfrac 8 {s^3} - \dfrac {3 s} {s^2 + 4} + \dfrac 5 {s + 1}$


Sources

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