Applied Mathematics/Laplace Transforms

The Laplace transform is an integral transform which is widely used in physics and engineering.

Laplace Transforms involve a technique to change an expression into another form that is easier to work with using an improper integral. We usually introduce Laplace Transforms in the context of differential equations, since we use them a lot to solve some differential equations that can't be solved using other standard techniques. However, Laplace Transforms require only improper integration techniques to use. So you may run across them in first year calculus.

Notation: The Laplace Transform is denoted as ${\displaystyle \displaystyle {\mathcal {L}}\left\{f(t)\right\}}$.

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace.

For a function ${\displaystyle f(t)}$, using Napier's constant ${\displaystyle e}$ and a complex number ${\displaystyle s}$, the Laplace transform ${\displaystyle F(s)}$ is defined as follows:

${\displaystyle F(s)={\mathcal {L}}\left\{f(t)\right\}(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt}$

The parameter ${\displaystyle s}$ is a complex number.

${\displaystyle s=\sigma +i\omega ,\,}$ with real numbers ${\displaystyle \sigma }$ and ${\displaystyle \omega }$.

This ${\displaystyle F(s)}$ is the Laplace transform of ${\displaystyle f(t)}$.

Here is what is going on.

Examples of Laplace transform

${\displaystyle f(t)}$	${\displaystyle F(s)={\mathcal {L}}\{f(t)\}}$
${\displaystyle C}$	${\displaystyle {\frac {C}{s}}}$
${\displaystyle t}$	${\displaystyle {\frac {1}{s^{2}}}}$
${\displaystyle t^{n}}$	${\displaystyle {\frac {n!}{s^{n+1}}}}$
${\displaystyle {\frac {t^{n-1}}{(n-1)!}}}$	${\displaystyle {\frac {1}{s^{n}}}}$
${\displaystyle e^{at}}$	${\displaystyle {\frac {1}{s-a}}}$
${\displaystyle e^{-at}}$	${\displaystyle {\frac {1}{s+a}}}$
${\displaystyle {\rm {cos}}\ \omega t}$	${\displaystyle {\frac {s}{s^{2}+{\omega }^{2}}}}$
${\displaystyle {\rm {sin}}\ \omega t}$	${\displaystyle {\frac {\omega }{s^{2}+{\omega }^{2}}}}$
${\displaystyle {\frac {t^{n-1}}{\Gamma (n)}}}$	${\displaystyle {\frac {1}{s^{n}}}}$ (n>0)
${\displaystyle \delta (t-a)}$	${\displaystyle e^{-as}}$
${\displaystyle H(t-a)}$	${\displaystyle {\frac {e^{-as}}{s}}}$

In the above table,

1. ${\displaystyle C}$ and ${\displaystyle a}$ are constants
2. ${\displaystyle n}$ is a natural number
3. ${\displaystyle \delta (t-a)}$ is the Delta function
4. ${\displaystyle H(t-a)}$ is the Heaviside function

ID	Function	Time domain ${\displaystyle x(t)={\mathcal {L}}^{-1}\left\{X(s)\right\}}$	Laplace domain ${\displaystyle X(s)={\mathcal {L}}\left\{x(t)\right\}}$	Region of convergence for causal systems
1	Ideal delay	${\displaystyle \delta (t-\tau )\ }$	${\displaystyle e^{-\tau s}\ }$
1a	Unit impulse	${\displaystyle \delta (t)\ }$	${\displaystyle 1\ }$	${\displaystyle \mathrm {all} \ s\,}$
2	Delayed nth power with frequency shift	${\displaystyle {\frac {(t-\tau )^{n}}{n!}}e^{-\alpha (t-\tau )}\cdot u(t-\tau )}$	${\displaystyle {\frac {e^{-\tau s}}{(s+\alpha )^{n+1}}}}$	${\displaystyle s>0\,}$
2a	nth Power	${\displaystyle {t^{n} \over n!}\cdot u(t)}$	${\displaystyle {1 \over s^{n+1}}}$	${\displaystyle s>0\,}$
2a.1	qth Power	${\displaystyle {t^{q} \over \Gamma (q+1)}\cdot u(t)}$	${\displaystyle {1 \over s^{q+1}}}$	${\displaystyle s>0\,}$
2a.2	Unit step	${\displaystyle u(t)\ }$	${\displaystyle {1 \over s}}$	${\displaystyle s>0\,}$
2b	Delayed unit step	${\displaystyle u(t-\tau )\ }$	${\displaystyle {e^{-\tau s} \over s}}$	${\displaystyle s>0\,}$
2c	Ramp	${\displaystyle t\cdot u(t)\ }$	${\displaystyle {\frac {1}{s^{2}}}}$	${\displaystyle s>0\,}$
2d	nth Power with frequency shift	${\displaystyle {\frac {t^{n}}{n!}}e^{-\alpha t}\cdot u(t)}$	${\displaystyle {\frac {1}{(s+\alpha )^{n+1}}}}$	${\displaystyle s>-\alpha \,}$
2d.1	Exponential decay	${\displaystyle e^{-\alpha t}\cdot u(t)\ }$	${\displaystyle {1 \over s+\alpha }}$	${\displaystyle s>-\alpha \ }$
3	Exponential approach	${\displaystyle (1-e^{-\alpha t})\cdot u(t)\ }$	${\displaystyle {\frac {\alpha }{s(s+\alpha )}}}$	${\displaystyle s>0\ }$
4	Sine	${\displaystyle \sin(\omega t)\cdot u(t)\ }$	${\displaystyle {\omega \over s^{2}+\omega ^{2}}}$	${\displaystyle s>0\ }$
5	Cosine	${\displaystyle \cos(\omega t)\cdot u(t)\ }$	${\displaystyle {s \over s^{2}+\omega ^{2}}}$	${\displaystyle s>0\ }$
6	Hyperbolic sine	${\displaystyle \sinh(\alpha t)\cdot u(t)\ }$	${\displaystyle {\alpha \over s^{2}-\alpha ^{2}}}$	${\displaystyle s>|\alpha |\ }$
7	Hyperbolic cosine	${\displaystyle \cosh(\alpha t)\cdot u(t)\ }$	${\displaystyle {s \over s^{2}-\alpha ^{2}}}$	${\displaystyle s>|\alpha |\ }$
8	Exponentially-decaying sine	${\displaystyle e^{-\alpha t}\sin(\omega t)\cdot u(t)\ }$	${\displaystyle {\omega \over (s+\alpha )^{2}+\omega ^{2}}}$	${\displaystyle s>-\alpha \ }$
9	Exponentially-decaying cosine	${\displaystyle e^{-\alpha t}\cos(\omega t)\cdot u(t)\ }$	${\displaystyle {s+\alpha \over (s+\alpha )^{2}+\omega ^{2}}}$	${\displaystyle s>-\alpha \ }$
10	nth Root	${\displaystyle {\sqrt[{n}]{t}}\cdot u(t)}$	${\displaystyle s^{-(n+1)/n}\cdot \Gamma \left(1+{\frac {1}{n}}\right)}$	${\displaystyle s>0\,}$
11	Natural logarithm	${\displaystyle \ln \left({t \over t_{0}}\right)\cdot u(t)}$	${\displaystyle -{t_{0} \over s}\ [\ \ln(t_{0}s)+\gamma \ ]}$	${\displaystyle s>0\,}$
12	Bessel function of the first kind, of order n	${\displaystyle J_{n}(\omega t)\cdot u(t)}$	${\displaystyle {\frac {\omega ^{n}\left(s+{\sqrt {s^{2}+\omega ^{2}}}\right)^{-n}}{\sqrt {s^{2}+\omega ^{2}}}}}$	${\displaystyle s>0\,}$ ${\displaystyle (n>-1)\,}$
13	Modified Bessel function of the first kind, of order n	${\displaystyle I_{n}(\omega t)\cdot u(t)}$	${\displaystyle {\frac {\omega ^{n}\left(s+{\sqrt {s^{2}-\omega ^{2}}}\right)^{-n}}{\sqrt {s^{2}-\omega ^{2}}}}}$	${\displaystyle s>|\omega |\,}$
14	Bessel function of the second kind, of order 0	${\displaystyle Y_{0}(\alpha t)\cdot u(t)}$
15	Modified Bessel function of the second kind, of order 0	${\displaystyle K_{0}(\alpha t)\cdot u(t)}$
16	Error function	${\displaystyle \mathrm {erf} (t)\cdot u(t)}$	${\displaystyle {e^{s^{2}/4}\operatorname {erfc} \left(s/2\right) \over s}}$	${\displaystyle s>0\,}$
17	Constant	${\displaystyle C}$	${\displaystyle \displaystyle {\frac {C}{s}}}$
Explanatory notes: ${\displaystyle u(t)\,}$ represents the Heaviside step function. ${\displaystyle \delta (t)\,}$ represents the Dirac delta function. ${\displaystyle \Gamma (z)\,}$ represents the Gamma function. ${\displaystyle \gamma \,}$ is the Euler-Mascheroni constant. ${\displaystyle t\,}$, a real number, typically represents time, although it can represent any independent dimension. ${\displaystyle s\,}$ is the complex angular frequency. ${\displaystyle \alpha \,}$, ${\displaystyle \beta \,}$, ${\displaystyle \tau \,}$, ${\displaystyle \omega \,}$ and ${\displaystyle C}$ are real numbers. ${\displaystyle n\,}$is an integer. A causal system is a system where the impulse response h(t) is zero for all time t prior to t = 0. In general, the ROC for causal systems is not the same as the ROC for anticausal systems. See also causality.

1. Calculate ${\displaystyle {\mathcal {L}}\{C\}}$ (where ${\displaystyle C}$ is a constant) using the integral definition.

${\displaystyle {\begin{array}{rcl}\displaystyle {\int _{0}^{\infty }e^{-st}C\,dt}&=&C\displaystyle {\int _{0}^{\infty }e^{-st}\,dt}\\&=&\displaystyle {C\lim _{b\to \infty }{\int _{0}^{b}e^{-st}\,dt}}\\&=&\displaystyle {\left.C\lim _{b\to \infty }{\frac {e^{-st}}{-s}}\right|_{t=0}^{t=b}}\\&=&\displaystyle {-{\frac {C}{s}}\left[\lim _{b\to \infty }{e^{-bs}}-e^{0}\right]}\\&=&\displaystyle {-{\frac {C}{s}}[0-1]}\\&=&\displaystyle {\frac {C}{s}}\end{array}}}$

${\displaystyle \therefore \displaystyle {\mathcal {L}}\left\{C\right\}={\frac {C}{s}}}$

2. Calculate ${\displaystyle {\mathcal {L}}\{e^{-at}\}}$ using the integral definition.

${\displaystyle {\begin{array}{rcl}\displaystyle \int _{0}^{\infty }e^{-st}\cdot e^{-at}\,dt&=&\displaystyle \int _{0}^{\infty }e^{-(s+a)t}\,dt\\&=&\displaystyle \lim _{b\to \infty }{\displaystyle \int _{0}^{b}e^{-(s+a)t}\,dt}\\&=&\displaystyle \lim _{b\to \infty }\left[\displaystyle {\frac {-e^{-(s+a)t}}{s+a}}\right]_{t=0}^{t=b}\\&=&\displaystyle \lim _{b\to \infty }\left[\displaystyle {\frac {-e^{-(s+a)b}}{s+a}}-\displaystyle {\frac {-e^{-(s+a)0}}{s+a}}\right]\\&=&\displaystyle {\frac {1}{s+a}}\end{array}}}$

${\displaystyle \therefore {\mathcal {L}}\{e^{-at}\}=\displaystyle {\frac {1}{s+a}}}$