Laplace Transform of Periodic Function/Proof 1
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Theorem
Let $f$ be periodic, that is:
- $\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$
Then:
- $\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \ds \int_0^T e^{-s t} \map f t \rd t$
where $\laptrans {\map f t}$ denotes the Laplace transform.
Proof
\(\ds \laptrans {\map f t}\) | \(=\) | \(\ds \int_0^{\infty} e^{-s t} \map f t \rd t\) | Definition of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^{\infty} \int_{k T}^{\paren {k + 1} T} e^{-s t} \map f t \rd t\) | Sum of Integrals on Adjacent Intervals for Integrable Functions: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^{\infty} \int_0^T e^{-s \paren {t + k T} } \map f {t + k T} \rd t\) | Change of Limits of Integration | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^{\infty} \int_0^T e^{-s \paren {t + k T} } \map f t \rd t\) | General Periodicity Property | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^{\infty} \int_0^T e^{-s t - s k T} \map f t \rd t\) | Real Multiplication Distributes over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^{\infty} \int_0^T e^{-s t} e^{- s k T} \map f t \rd t\) | Product of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^{\infty} e^{- s k T} \int_0^T e^{-s t} \map f t \rd t\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\int_0^T e^{-s t} \map f t \rd t} \sum_{k \mathop = 0}^{\infty} e^{- s k T}\) | Scaling of Summations | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\int_0^T e^{-s t} \map f t \rd t} \sum_{k \mathop = 0}^{\infty} \paren {e^{- s T} }^k\) | Power of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\int_0^T e^{-s t} \map f t \rd t} \paren {\frac 1 {1 - e^{-s T} } }\) | Sum of Infinite Geometric Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {1 - e^{-s T} } \int_0^T e^{-s t} \map f t \rd t\) | Real Multiplication is Commutative |
$\blacksquare$