Laplace Transform of Periodic Function

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Theorem

Let $f: \R \to \R$ be a real function.

Let $f$ be periodic, that is:

$\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$


Then:

$\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \displaystyle \int_0^T e^{-s t} \map f t \rd t$

where $\laptrans {\map f t}$ denotes the Laplace transform.


Proof 1

\(\displaystyle \laptrans {\map f t}\) \(=\) \(\displaystyle \int_0^{\infty} e^{-s t} \map f t \rd t\) Definition of Laplace Transform
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^{\infty} \int_{k T}^{\paren {k + 1} T} e^{-s t} \map f t \rd t\) Sum of Integrals on Adjacent Intervals for Integrable Functions: Corollary
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^{\infty} \int_0^T e^{-s \paren {t + k T} } \map f {t + k T} \rd t\) Change of Limits of Integration
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^{\infty} \int_0^T e^{-s \paren {t + k T} } \map f t \rd t\) General Periodicity Property
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^{\infty} \int_0^T e^{-s t - s k T} \map f t \rd t\) Real Multiplication Distributes over Addition
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^{\infty} \int_0^T e^{-s t} e^{- s k T} \map f t \rd t\) Exponent Combination Laws: Product of Powers
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^{\infty} e^{- s k T} \int_0^T e^{-s t} \map f t \rd t\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \paren {\int_0^T e^{-s t} \map f t \rd t} \sum_{k \mathop = 0}^{\infty} e^{- s k T}\) Scaling of Summations
\(\displaystyle \) \(=\) \(\displaystyle \paren {\int_0^T e^{-s t} \map f t \rd t} \sum_{k \mathop = 0}^{\infty} \paren {e^{- s T} }^k\) Exponent Combination Laws: Power of Power
\(\displaystyle \) \(=\) \(\displaystyle \paren {\int_0^T e^{-s t} \map f t \rd t} \paren {\frac 1 {1 - e^{-s T} } }\) Sum of Infinite Geometric Progression
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {1 - e^{-s T} } \int_0^T e^{-s t} \map f t \rd t\) Real Multiplication is Commutative

$\blacksquare$


Proof 2

\(\displaystyle \laptrans {\map f t}\) \(=\) \(\displaystyle \int_0^{\infty} e^{-s t} \map f t \rd t\) Definition of Laplace Transform
\(\displaystyle \) \(=\) \(\displaystyle \int_0^T e^{-s t} \map f t \rd t + \int_T^{\infty} e^{-s t} \map f t \rd t\) Sum of Integrals on Adjacent Intervals for Integrable Functions
\(\displaystyle \) \(=\) \(\displaystyle \int_0^T e^{-s t} \map f t \rd t + \int_0^{\infty} e^{-s \paren {t + T} } \map f {t + T} \rd t\) Change of Limits of Integration
\(\displaystyle \) \(=\) \(\displaystyle \int_0^T e^{-s t} \map f t \rd t + \int_0^{\infty} e^{-s \paren {t + T} } \map f t \rd t\) Definition of Real Periodic Function
\(\displaystyle \) \(=\) \(\displaystyle \int_0^T e^{-s t} \map f t \rd t + \int_0^{\infty} e^{-s t - s T} \map f t \rd t\) Real Multiplication Distributes over Addition
\(\displaystyle \) \(=\) \(\displaystyle \int_0^T e^{-s t} \map f t \rd t + \int_0^{\infty} e^{-s t} e^{-s T} \map f t \rd t\) Exponent Combination Laws: Product of Powers
\(\displaystyle \) \(=\) \(\displaystyle \int_0^T e^{-s t} \map f t \rd t + e^{-s T} \int_0^{\infty} e^{-s t} \map f t \rd t\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \int_0^T e^{-s t} \map f t \rd t + e^{-s T} \laptrans {\map f t}\) Definition of Laplace Transform
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {1 - e^{-s T} } \laptrans {\map f t}\) \(=\) \(\displaystyle \int_0^T e^{-s t} \map f t \rd t\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \laptrans {\map f t}\) \(=\) \(\displaystyle \frac 1 {1 - e^{-s T} } \int_0^T e^{-s t} \map f t \rd t\)

$\blacksquare$


Proof 3

\(\displaystyle \laptrans {\map f t}\) \(=\) \(\displaystyle \int_0^{\infty} e^{-s t} \map f t \rd t\) Definition of Laplace Transform
\(\displaystyle \) \(=\) \(\displaystyle \int_0^T e^{-s t} \map f t \rd t + \int_T^{2 T} e^{-s t} \map f t \rd t + \int_{2 T}^{3 T} e^{-s t} \map f t \rd t + \dotsb\) Sum of Integrals on Adjacent Intervals for Integrable Functions
\(\displaystyle \) \(=\) \(\displaystyle \int_0^T e^{-s u} \map f u \rd t + \int_T^{2 T} e^{-s \paren {u + T} } \map f {u + T} \rd u + \int_{2 T}^{3 T} e^{-s \paren {u + 2 T} } \map f {u + 2 T} \rd u + \dotsb\) Integration by Substitution: $t = u$, $t = u + T$, $t = u + 2 T$, $\ldots$
\(\displaystyle \) \(=\) \(\displaystyle \int_0^T e^{-s u} \map f u \rd t + e^{-s T} \int_0^T e^{-s u} \map f u \rd u + e^{-2 s T} \int_0^T e^{-s u} \map f u \rd u + \dotsb\) Laplace Transform of Function of t minus a, and adjusting limits of integration
\(\displaystyle \) \(=\) \(\displaystyle \paren {1 + e^{-s T} + e^{-2 s T} + \dotsb} \int_0^T e^{-s u} \map f u \rd u\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {1 - e^{-s T} } \int_0^T e^{-s t} \map f t \rd t\) Sum of Infinite Geometric Progression

$\blacksquare$


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