# Laplace Transform of Identity Mapping

## Theorem

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

Let $\map {I_\R} t$ denote the identity mapping on $\R$ for $t > 0$.

Then:

$\laptrans {\map {I_\R} t} = \dfrac 1 {s^2}$

for $\map \Re s > 0$.

## Proof 1

 $\displaystyle \laptrans {\map {I_\R} t}$ $=$ $\displaystyle \laptrans t$ Definition of Identity Mapping $\displaystyle$ $=$ $\displaystyle \int_0^{\to +\infty} t e^{-s t} \rd t$ Definition of Laplace Transform $\displaystyle$ $=$ $\displaystyle \intlimits {\frac {e^{-s t} } {-s} \paren {t - \frac 1 {-s} } } {t \mathop = 0} {t \mathop \to +\infty}$ Primitive of $x e^{a x}$ $\displaystyle$ $=$ $\displaystyle -\frac 1 s \lim_{t \mathop \to +\infty} \frac t { e^{s t} } - \paren {0 - \frac 1 {s^2} }$ Exponential of Zero and One, Exponent Combination Laws: Negative Power $\displaystyle$ $=$ $\displaystyle 0 + \frac 1 {s^2}$ Limit at Infinity of Polynomial over Complex Exponential $\displaystyle$ $=$ $\displaystyle \frac 1 {s^2}$

$\blacksquare$

## Proof 2

 $\displaystyle \laptrans {\map {I_\R} t}$ $=$ $\displaystyle \laptrans t$ Definition of Identity Mapping $\displaystyle$ $=$ $\displaystyle \int_0^{\to +\infty} t e^{-st} \rd t$ Definition of Laplace Transform

From Integration by Parts:

$\displaystyle \int f g' \rd t = f g - \int f'g \rd t$

Here:

 $\displaystyle f$ $=$ $\displaystyle t$ $\displaystyle \leadsto \ \$ $\displaystyle f'$ $=$ $\displaystyle 1$ Derivative of Identity Function $\displaystyle g'$ $=$ $\displaystyle e^{-st}$ $\displaystyle \leadsto \ \$ $\displaystyle g$ $=$ $\displaystyle -\frac 1 s e^{-s t}$ Primitive of Exponential Function

So:

 $\displaystyle \int t e^{-s t} \rd t$ $=$ $\displaystyle -\frac t s e^{-s t} - \frac 1 s \int e^{-s t} \rd t$ $\displaystyle$ $=$ $\displaystyle -\frac t s e^{-s t} - \frac 1 {s^2} e^{-s t}$ Primitive of Exponential Function

Evaluating at $t = 0$ and $t \to +\infty$:

 $\displaystyle \laptrans t$ $=$ $\displaystyle \intlimits {-\frac t s e^{-s t} - \frac 1 {s^2} e^{-s t} } {t \mathop = 0} {t \mathop \to +\infty}$ $\displaystyle$ $=$ $\displaystyle -\frac 1 s \lim_{t \mathop \to +\infty} \frac t { e^{s t} } - \paren {0 - \frac 1 {s^2} }$ Exponential of Zero and One, Exponent Combination Laws: Negative Power $\displaystyle$ $=$ $\displaystyle 0 + \frac 1 {s^2}$ Limit at Infinity of Polynomial over Complex Exponential $\displaystyle$ $=$ $\displaystyle \frac 1 {s^2}$

$\blacksquare$

## Proof 3

$(1): \quad \laptrans {\map {I_\R'} t} = s \laptrans {\map {I_\R} t} - \map {I_\R} 0$

under suitable conditions.

Then:

 $\displaystyle \map {I_\R} t$ $=$ $\displaystyle t$ $\displaystyle \leadsto \ \$ $\displaystyle \map {I_\R'} t$ $=$ $\displaystyle 1$ $\displaystyle \map {I_\R} 0$ $=$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle \laptrans 1$ $=$ $\displaystyle \dfrac 1 s$ Laplace Transform of 1 $\displaystyle$ $=$ $\displaystyle s \laptrans {\map {I_\R} t} - 0$ from $(1)$, substituting for $\map f t$ and $\map f 0$ $\displaystyle \leadsto \ \$ $\displaystyle s \laptrans {\map {I_\R} t}$ $=$ $\displaystyle \dfrac 1 s$ $\displaystyle \leadsto \ \$ $\displaystyle \laptrans {\map {I_\R} t}$ $=$ $\displaystyle \dfrac 1 {s^2}$

$\blacksquare$