# Laplace Transform of Periodic Function

## 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:

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

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

## Proof 1

 $\ds \laptrans {\map f t}$ $=$ $\ds \int_0^{\infty} e^{-s t} \map f t \rd t$ 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

$\blacksquare$

## Proof 2

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

$\blacksquare$

## Proof 3

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

$\blacksquare$