拉普拉斯轉換（Laplace Transform）

Laurent series

The Laurent series for a complex function $f(z)$ about a point $c$ is given by

\[f(z)={\textstyle\textstyle\mbox{$Σ$}_{n=-\infty}^\infty}{\textstyle a_n}{\textstyle{(z-c)}^n}\]

where $a_n$ and $c$ are constants, with $a_n$ defined by a contour integral that generalizes Cauchy's integral formula:

\[{\textstyle{\scriptstyle a}_n}{\textstyle=}\frac1{2\mathrm{πi}}{\textstyle{\scriptstyle\oint}_\gamma}\frac{f(z)}{{(z-c)}^{n+1}}{\textstyle d}{\textstyle z}\]

Suppose that $f(z)$ is analytic on the annulus

\[A: r_1 < |z - z_0| < r_2. \nonumber\]

Then $f(z)$ can be expressed as a series

\[f(z) = \sum_{n = 1}^{\infty} \dfrac{b_n}{(z - z_0)^n} + \sum_{n = 0}^{\infty} a_n (z - z_0)^n. \nonumber\]

The coefficients have the formulus

\[\begin{array} {l} {a_n = \dfrac{1}{2\pi i} \int_{\gamma} \dfrac{f(w)}{(w - z_0)^{n + 1}}\ dw} \\ {b_n = \dfrac{1}{2\pi i} \int_{\gamma} f(w) (w - z_0)^{n - 1}\ dw} \end{array} \nonumber\]

where $γ$ is any circle $|w−z_0|=r$ inside the annulus, i.e. $\;r_1<\;r\;<\;r_2 $

Furthermore

The series $\sum_{n = 0}^{\infty} a_n (z - z_0)^n$ converges to an analytic function for $|z - z_0| < r_2$

The series $\sum_{n = 1}^{\infty} \dfrac{b_n}{(z - z_0)^n}$ converges to an analytic function for $|z - z_0| > r_1$

Together, the series both converge on the annulus $A$ where $f$ is analytic.

Conformal mapping

Signals and Systems

Signals ultilize "Fourier Transform"

Systems make use of "Laplace Transform"

Fourier transform time domain is $from\;-\infty\;to\;\infty$

Laplace transform time domain is $from\;0^-\;to\;\infty$

The Laplace transform converts a signal to a complex plane. The Fourier transform transforms the same signal into the jw plane and is a subset of the Laplace transform in which the real part is 0.

Discrete Fourier Transform - Represents the magnitude and phase of a discrete system.

Laplace Transforms - Represents the magnitude, phase, and dampening of a continuous system.

Z Transform - Represents the magnitude, phase, and dampening of a discrete system.

Laplace Transform

拉普拉斯轉換

The Laplace transform is used where the Fourier transform cannot be used. The Laplace transform redefines the transform and includes an exponential convergence factor σ along with jω. Therefore, using the Laplace transform the time-domain signal x(t) can be represented as a sum of complex exponential functions of the form $e^{st}$.

The advantage of using the Laplace transform is that it converts an ODE (Ordinary differential equation) into an algebraic equation of the same order that is simpler to solve, even though it is a function of a complex variable. The chapter discusses ways of solving ODEs using the phasor notation for sinusoidal signals.

In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace, is an integral transform that converts a function of a real variable (usually $t$, in the time domain) to a function of a complex variable $s$ (in the complex-valued frequency domain, also known as s-domain, or s-plane).

The Laplace transform is defined (for suitable functions $f$) by the integral:

$ℒ\lbrack f(t)\rbrack=F(s) \overset{\rm{def}}{=} \int_0^{\infty} e^{-st}f(t)\,dt. \nonumber$

Table of Laplace Transforms

	$f\left( t \right) = {\mathcal{L}^{\,\, - 1}}\left\{ {F\left( s \right)} \right\}$	$F\left( s \right) = \mathcal{L}\left\{ {f\left( t \right)} \right\}$
1.	1	$\displaystyle \frac{1}{s}$
2.	${{\bf{e}}^{a\,t}}$	$\displaystyle \frac{1}{{s - a}}$
3.	${t^n},\,\,\,\,\,n = 1,2,3, \ldots $	$\displaystyle \frac{{n!}}{{{s^{n + 1}}}}$
4.	${t^p}$, $p > -1$	$\displaystyle \frac{{\Gamma \left( {p + 1} \right)}}{{{s^{p + 1}}}}$
5.	$\sqrt t $	$\displaystyle \frac{{\sqrt \pi }}{{2{s^{\frac{3}{2}}}}}$
6.	${t^{n - \frac{1}{2}}},\,\,\,\,\,n = 1,2,3, \ldots $	$\displaystyle \frac{{1 \cdot 3 \cdot 5 \cdots \left( {2n - 1} \right)\sqrt \pi }}{{{2^n}{s^{n + \frac{1}{2}}}}}$
7.	$\sin \left( {at} \right)$	$\displaystyle \frac{a}{{{s^2} + {a^2}}}$
8.	$\cos \left( {at} \right)$	$\displaystyle \frac{s}{{{s^2} + {a^2}}}$
9.	$t\sin \left( {at} \right)$	$\displaystyle \frac{{2as}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}$
10.	$t\cos \left( {at} \right)$	$\displaystyle \frac{{{s^2} - {a^2}}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}$
11.	$\sin \left( {at} \right) - at\cos \left( {at} \right)$	$\displaystyle \frac{{2{a^3}}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}$
12.	$\sin \left( {at} \right) + at\cos \left( {at} \right)$	$\displaystyle \frac{{2a{s^2}}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}$
13.	$\cos \left( {at} \right) - at\sin \left( {at} \right)$	$\displaystyle \frac{{s\left( {{s^2} - {a^2}} \right)}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}$
14.	$\cos \left( {at} \right) + at\sin \left( {at} \right)$	$\displaystyle \frac{{s\left( {{s^2} + 3{a^2}} \right)}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}$
15.	$\sin \left( {at + b} \right)$	$\displaystyle \frac{{s\sin \left( b \right) + a\cos \left( b \right)}}{{{s^2} + {a^2}}}$
16.	$\cos \left( {at + b} \right)$	$\displaystyle \frac{{s\cos \left( b \right) - a\sin \left( b \right)}}{{{s^2} + {a^2}}}$
17.	$\sinh \left( {at} \right)$	$\displaystyle \frac{a}{{{s^2} - {a^2}}}$
18.	$\cosh \left( {at} \right)$	$\displaystyle \frac{s}{{{s^2} - {a^2}}}$
19.	${{\bf{e}}^{at}}\sin \left( {bt} \right)$	$\displaystyle \frac{b}{{{{\left( {s - a} \right)}^2} + {b^2}}}$
20.	${{\bf{e}}^{at}}\cos \left( {bt} \right)$	$\displaystyle \frac{{s - a}}{{{{\left( {s - a} \right)}^2} + {b^2}}}$
21.	${{\bf{e}}^{at}}\sinh \left( {bt} \right)$	$\displaystyle \frac{b}{{{{\left( {s - a} \right)}^2} - {b^2}}}$
22.	${{\bf{e}}^{at}}\cosh \left( {bt} \right)$	$\displaystyle \frac{{s - a}}{{{{\left( {s - a} \right)}^2} - {b^2}}}$
23.	${t^n}{{\bf{e}}^{at}},\,\,\,\,\,n = 1,2,3, \ldots $	$\displaystyle \frac{{n!}}{{{{\left( {s - a} \right)}^{n + 1}}}}$
24.	$f\left( {ct} \right)$	$\displaystyle \frac{1}{c}F\left( {\frac{s}{c}} \right)$
25.	${u_c}\left( t \right) = u\left( {t - c} \right)$	$\displaystyle \frac{{{{\bf{e}}^{ - cs}}}}{s}$
26.	$\delta \left( {t - c} \right)$	${{\bf{e}}^{ - cs}}$
27.	${u_c}\left( t \right)f\left( {t - c} \right)$	${{\bf{e}}^{ - cs}}F\left( s \right)$
28.	${u_c}\left( t \right)g\left( t \right)$	${{\bf{e}}^{ - cs}}{\mathcal{L}}\left\{ {g\left( {t + c} \right)} \right\}$
29.	${{\bf{e}}^{ct}}f\left( t \right)$	$F\left( {s - c} \right)$
30.	${t^n}f\left( t \right),\,\,\,\,\,n = 1,2,3, \ldots $	${\left( { - 1} \right)^n}{F^{\left( n \right)}}\left( s \right)$
31.	$\displaystyle \frac{1}{t}f\left( t \right)$	$\int_{{\,s}}^{{\,\infty }}{{F\left( u \right)\,du}}$
32.	$\displaystyle \int_{{\,0}}^{{\,t}}{{\,f\left( v \right)\,dv}}$	$\displaystyle \frac{{F\left( s \right)}}{s}$
33.	$\displaystyle \int_{{\,0}}^{{\,t}}{{f\left( {t - \tau } \right)g\left( \tau \right)\,d\tau }}$	$F\left( s \right)G\left( s \right)$
34.	$f\left( {t + T} \right) = f\left( t \right)$	$\displaystyle \frac{{\displaystyle \int_{{\,0}}^{{\,T}}{{{{\bf{e}}^{ - st}}f\left( t \right)\,dt}}}}{{1 - {{\bf{e}}^{ - sT}}}}$
35.	$f'\left( t \right)$	$sF\left( s \right) - f\left( 0 \right)$
36.	$f''\left( t \right)$	${s^2}F\left( s \right) - sf\left( 0 \right) - f'\left( 0 \right)$
37.	${f^{\left( n \right)}}\left( t \right)$	${s^n}F\left( s \right) - {s^{n - 1}}f\left( 0 \right) - {s^{n - 2}}f'\left( 0 \right) \cdots - s{f^{\left( {n - 2} \right)}}\left( 0 \right) - {f^{\left( {n - 1} \right)}}\left( 0 \right)$

Laplace Transform

Selected Laplace transforms

The following functions and variables are used in the table below:

δ represents the Dirac delta function.
u(t) represents the Heaviside step function. Literature may refer to this by other notation, including $1(t)$ or $H(t)$.
Γ(z) represents the Gamma function.
γ is the Euler–Mascheroni constant.
t is a real number. It typically represents time, although it can represent any independent dimension.
s is the complex frequency domain parameter, and Re(s) is its real part.
n is an integer.
α, τ, and ω are real numbers.
q is a complex number.

Function	Time domain $f(t)={\mathcal L}^{-1}\{F(s)\}$	Laplace s-domain $F(s)={\mathcal L}\{f(t)\}$	Region of convergence	Reference
unit impulse	\[\delta(t)\]	\[1\]	all s	inspection
delayed impulse	\[\delta(t-\tau)\]	\[e^{-\tau s}\]	Re(s) > 0	time shift of unit impulse
unit step	\[u(t)\]	\[\frac1s\]	Re(s) > 0	integrate unit impulse
delayed unit step	\[u(t-\tau)\]	\[{\frac1s}e^{-\tau s}\]	Re(s) > 0	time shift of unit step
ramp	\[t⋅u(t)\]	\[\frac1{s^2}\]	Re(s) > 0	integrate unit impulse twice
nth power (for integer n)	\[t^n⋅u(t)\]	\[\frac{n!}{s^{n+1}}\]	Re(s) > 0 (n > −1)	Integrate unit step n times
qth power (for complex q)	\[t^q⋅u(t)\]	\[\frac{{\mathrm\Gamma}(q+1)}{s^{q+1}}\]	Re(s) > 0 Re(q) > −1
nth root	\[{\sqrt[n]t}⋅u(t)\]	\[{\frac1{s^{\frac1n+1}}}{\mathrm\Gamma}{({\frac1n}+1)}\]	Re(s) > 0	Set q = 1/n above.
nth power with frequency shift	\[t^ne^{-\alpha t}⋅u(t)\]	\[\frac{n!}{(s+\alpha)^{n+1}}\]	Re(s) > −α	Integrate unit step, apply frequency shift
delayed nth power with frequency shift	\[(t-\tau)^ne^{-\alpha(t-\tau)}⋅u(t-\tau)\]	\[\frac{n!⋅e^{-\tau s}}{(s+\alpha)^{n+1}}\]	Re(s) > −α	Integrate unit step, apply frequency shift, apply time shift
exponential decay	\[e^{-\alpha t}u(t)\]	\[\frac1{s+\alpha}\]	Re(s) > −α	Frequency shift of unit step
two-sided exponential decay (only for bilateral transform)	\[e^{-\alpha\vert t\vert}\]	\[\frac{2\alpha}{\alpha^2-s^2}\]	−α < Re(s) < α	Frequency shift of unit step
exponential approach	\[(1-e^{-\alpha t})⋅u(t)\]	\[\frac\alpha{s(s+\alpha)}\]	Re(s) > 0	Unit step minus exponential decay
sine	\[\sin(\omega t)⋅u(t)\]	\[\frac\omega{s^2+\omega^2}\]	Re(s) > 0
cosine	\[\cos(\omega t)⋅u(t)\]	\[\frac s{s^2+\omega^2}\]	Re(s) > 0
hyperbolic sine	\[\sinh(\alpha t)⋅u(t)\]	\[\frac\alpha{s^2-\alpha^2}\]	Re(s) > \|α\|
hyperbolic cosine	\[\cosh(\alpha t)⋅u(t)\]	\[\frac s{s^2-\alpha^2}\]	Re(s) > \|α\|
exponentially decaying sine wave	\[e^{-\alpha t}\sin(\omega t)⋅u(t)\]	\[\frac\omega{(s+\alpha)^2+\omega^2}\]	Re(s) > −α
exponentially decaying cosine wave	\[e^{-\alpha t}\cos(\omega t)⋅u(t)\]	\[\frac{s+\alpha}{(s+\alpha)^2+\omega^2}\]	Re(s) > −α
natural logarithm	\[\ln(t)⋅u(t)\]	\[\frac{-\ln(s)-\gamma}s\]	Re(s) > 0
Bessel function of the first kind, of order n	\[J_n(\omega t)⋅u(t)\]	\[\frac{{({\sqrt{s^2+\omega^2}}-s)}^{{}n}}{\omega^n{\sqrt{s^2+\omega^2}}}\]	Re(s) > 0 (n > −1)
Error function	\[\mathrm{erf}(t)⋅u(t)\]	\[{\frac{e^{s^2/4}}s}{}{(1-\mathrm{erf}{({\frac s2})})}\]	Re(s) > 0

Region of convergence ROC

Laplace transform

Region of Convergence (ROC) is defined as the set of points in s-plane for which the Laplace transform of a function x(t) converges.

If x(t) is absolutely integral and it is of finite duration, then ROC is entire s-plane.
If x(t) is a right sided sequence then ROC : Re{s} > σo.
If x(t) is a left sided sequence then ROC : Re{s} < σo.
If x(t) is a two sided sequence then ROC is the combination of two regions.

$$f(t)$$	$$F(s)$$	ROC
$$ u(t) $$	$${1\over s}$$	Re{s} > 0
$$ t\, u(t) $$	$${1\over s^2} $$	Re{s} > 0
$$ t^n\, u(t) $$	$$ {n! \over s^{n+1}} $$	Re{s} > 0
$$ e^{at}\, u(t) $$	$$ {1\over s-a} $$	Re{s} > a
$$ e^{-at}\, u(t) $$	$$ {1\over s+a} $$	Re{s} > -a
$$ e^{at}\, u(t) $$	$$ - {1\over s-a} $$	Re{s} < a
$$ e^{-at}\, u(-t) $$	$$ - {1\over s+a} $$	Re{s} < -a
$$ t\, e^{at}\, u(t) $$	$$ {1 \over (s-a)^2} $$	Re{s} > a
$$ t^{n} e^{at}\, u(t) $$	$$ {n! \over (s-a)^{n+1}} $$	Re{s} > a
$$ t\, e^{-at}\, u(t) $$	$$ {1 \over (s+a)^2} $$	Re{s} > -a
$$ t^n\, e^{-at}\, u(t) $$	$${n! \over (s+a)^{n+1}} $$	Re{s} > -a
$$ t\, e^{at}\, u(-t) $$	$$ - {1 \over (s-a)^2} $$	Re{s} < a
$$ t^n\, e^{at}\, u(-t) $$	$$ - {n! \over (s-a)^{n+1}} $$	Re{s} < a
$$ t\, e^{-at}\,u(-t) $$	$$ - {1 \over (s+a)^2} $$	Re{s} < -a
$$ t^n\, e^{-at}\, u(-t) $$	$$ - {n! \over (s+a)^{n+1}} $$	Re{s} < -a
$$ e^{-at} \cos \, bt $$	$$ {s+a \over (s+a)^2 + b^2 } $$
$$ e^{-at} \sin\, bt $$	$$ {b \over (s+a)^2 + b^2 } $$

Poles and Zeros

Inverse Laplace transform

In mathematics, the inverse Laplace transform of a function $F(s)$ is the piecewise-continuous and exponentially-restricted real function $f(t)$ which has the property:

$ℒ\{f(t)\}=F(s)$

$f(t)=ℒ^{-1}\{F(s)\}$

where $ℒ$ denotes the Laplace transform.

The Laplace transform and the inverse Laplace transform together have a number of properties that make them useful for analysing linear dynamical systems.

Mellin's inverse formula

An integral formula for the inverse Laplace transform, called the Mellin's inverse formula, the Bromwich integral, or the Fourier–Mellin integral, is given by the line integral:

$f(t)=ℒ^{-1}\{F(s)\}=\frac1{2\pi i}\lim_{T\rightarrow\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}F(s)ds$

Laplace Transform Intergral and Derivative

Laplace transform of a derivative

$ℒ(t^n)(s)=\frac{n}{s}{ℒ(t^{n-1})}$

$sℒ(t^n)(s)= ℒ{(nt^{n-1})}$

$sℒ(t^n)(s)= ℒ\{(t^{n})'\}$

$sℒ(t^n)(s)-0= ℒ\{(t^{n})'\}$

$ℒ\{y'\}=sℒ(y)-y(0)$

Laplace Transform of Higher Order Derivatives

$ℒ\{f^{(n)}(t)\}=s^nℒ\{f(t)\}-{\textstyle\sum_{j=1}^n}s^{j-1}f^{(n-j)}(0)$

Laplace transforms of an integral

$if\;G(s)=ℒ\{g(t)\},\;then\;ℒ\{\int_0^tg(t)dt\}=\frac{G(s)}s$

the value of the integral when $t=0$

$ℒ\{\int_0^tg(t)dt\}=\frac{G(s)}s+\frac1s{\left[\int g(t)dt\right]}_{t=0}$

Final Value Theorem of Laplace Transform

\[\lim_{t \rightarrow \infty}\mathit{x}\mathrm{(\mathit{t})}\:\mathrm{=}\:\mathit{x}\mathrm{(\infty)}\:\mathrm{=}\:\lim_{s \rightarrow 0}\mathit{sX}\mathrm{(\mathit{s})}\]

$\lim_{s \rightarrow 0}[\int_{\mathrm{0}}^{\infty}\mathit{\frac{\mathit{dx}\mathrm{(\mathit{t})}}{\mathit{dt}}\mathit{e^{-st}}}\:\mathit{dt}]\:\mathrm{=}\:\lim_{s \rightarrow \infty}[\mathit{sX}\mathrm{(\mathit{s})}-\mathit{x}\mathrm{(0^-)}]$

$\Rightarrow\int_{\mathrm{0}}^{\infty}\mathit{\frac{\mathit{dx}\mathrm{(\mathit{t})}}{\mathit{dt}}}\:\mathit{dt}\:\mathrm{=}\:\lim_{s \rightarrow 0}[\mathit{sX}\mathrm{(\mathit{s})}-\mathit{x}\mathrm{(0^-)}]$

$\Rightarrow[\mathit{x}\mathrm{(\mathit{t})}]_0^\infty\:\mathrm{=}\:\lim_{s \rightarrow 0}[\mathit{sX}\mathrm{(s)}-\mathit{x}\mathrm{(0^-)}]$

$\Rightarrow\mathit{x}\mathrm{(\infty)}-\mathit{x}\mathrm{(0^-)}\:\mathrm{=}\:\lim_{s \rightarrow 0}[\mathit{sX}\mathrm{(\mathit{s})-\mathit{x}\mathrm{(0^-)}}]$

$⇒x(∞)=lims→0sX(s)$

Therefore, we have,

$\lim_{t \rightarrow \infty}\mathit{x}\mathrm{(\mathit{t})}\:\mathrm{=}\:\mathit{x}\mathrm{(\infty)}\:\mathrm{=}\:\lim_{s \rightarrow 0}\mathit{sX}\mathrm{(\mathit{s})}$

Property proofs

The one-sided Laplace transform is defined as

\[ℒ\left\{\,f(t)\,\right\}=F(s)=\int_{0^-}^\infty e^{-st}f(t)\;\mathrm dt\]

Linearity Property

$ℒ \left\{ a \, f(t) + b \, g(t) \right\} = a \, ℒ \left\{ f(t) \right\} + b \, ℒ \left\{ g(t) \right\}$

$ℒ \left\{ a \, f(t) + b \, g(t) \right\} = \int_0^\infty e^{-st}\left[ a \, f(t) + b \, g(t) \right] \, dt$

$ℒ \left\{ a \, f(t) + b \, g(t) \right\} = a\int_0^\infty e^{-st} f(t) \, dt + b\int_0^\infty e^{-st} g(t) \, dt$

$ℒ \left\{ a \, f(t) + b \, g(t) \right\} = a \, ℒ \left\{ f(t) \right\} + b \, ℒ \left\{ g(t) \right\}$

Shifting Property

if $ℒ \left\{ f(t) \right\} = F(s)$, when $s > a$ then,

$ℒ \left\{ e^{at} \, f(t) \right\} = F(s - a)$

$F(s) = \int_0^\infty e^{-st} f(t) \, dt$

$F(s - a) = \int_0^\infty e^{-(s - a)t} f(t) \, dt$

$F(s - a) = \int_0^\infty e^{-st + at} f(t) \, dt$

$F(s - a) = \int_0^\infty e^{-st} e^{at} f(t) \, dt$

$F(s - a) = ℒ \left\{ e^{at} f(t) \right\}$

Shifting Property

	\(f\left( t \right) = {\mathcal{L}^{\,\, - 1}}\left\{ {F\left( s \right)} \right\}\)	\(F\left( s \right) = \mathcal{L}\left\{ {f\left( t \right)} \right\}\)
1.	1	\(\displaystyle \frac{1}{s}\)
2.	\({{\bf{e}}^{a\,t}}\)	\(\displaystyle \frac{1}{{s - a}}\)
3.	\({t^n},\,\,\,\,\,n = 1,2,3, \ldots \)	\(\displaystyle \frac{{n!}}{{{s^{n + 1}}}}\)
4.	\({t^p}\), \(p > -1\)	\(\displaystyle \frac{{\Gamma \left( {p + 1} \right)}}{{{s^{p + 1}}}}\)
5.	\(\sqrt t \)	\(\displaystyle \frac{{\sqrt \pi }}{{2{s^{\frac{3}{2}}}}}\)
6.	\({t^{n - \frac{1}{2}}},\,\,\,\,\,n = 1,2,3, \ldots \)	\(\displaystyle \frac{{1 \cdot 3 \cdot 5 \cdots \left( {2n - 1} \right)\sqrt \pi }}{{{2^n}{s^{n + \frac{1}{2}}}}}\)
7.	\(\sin \left( {at} \right)\)	\(\displaystyle \frac{a}{{{s^2} + {a^2}}}\)
8.	\(\cos \left( {at} \right)\)	\(\displaystyle \frac{s}{{{s^2} + {a^2}}}\)
9.	\(t\sin \left( {at} \right)\)	\(\displaystyle \frac{{2as}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}\)
10.	\(t\cos \left( {at} \right)\)	\(\displaystyle \frac{{{s^2} - {a^2}}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}\)
11.	\(\sin \left( {at} \right) - at\cos \left( {at} \right)\)	\(\displaystyle \frac{{2{a^3}}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}\)
12.	\(\sin \left( {at} \right) + at\cos \left( {at} \right)\)	\(\displaystyle \frac{{2a{s^2}}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}\)
13.	\(\cos \left( {at} \right) - at\sin \left( {at} \right)\)	\(\displaystyle \frac{{s\left( {{s^2} - {a^2}} \right)}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}\)
14.	\(\cos \left( {at} \right) + at\sin \left( {at} \right)\)	\(\displaystyle \frac{{s\left( {{s^2} + 3{a^2}} \right)}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}\)
15.	\(\sin \left( {at + b} \right)\)	\(\displaystyle \frac{{s\sin \left( b \right) + a\cos \left( b \right)}}{{{s^2} + {a^2}}}\)
16.	\(\cos \left( {at + b} \right)\)	\(\displaystyle \frac{{s\cos \left( b \right) - a\sin \left( b \right)}}{{{s^2} + {a^2}}}\)
17.	\(\sinh \left( {at} \right)\)	\(\displaystyle \frac{a}{{{s^2} - {a^2}}}\)
18.	\(\cosh \left( {at} \right)\)	\(\displaystyle \frac{s}{{{s^2} - {a^2}}}\)
19.	\({{\bf{e}}^{at}}\sin \left( {bt} \right)\)	\(\displaystyle \frac{b}{{{{\left( {s - a} \right)}^2} + {b^2}}}\)
20.	\({{\bf{e}}^{at}}\cos \left( {bt} \right)\)	\(\displaystyle \frac{{s - a}}{{{{\left( {s - a} \right)}^2} + {b^2}}}\)
21.	\({{\bf{e}}^{at}}\sinh \left( {bt} \right)\)	\(\displaystyle \frac{b}{{{{\left( {s - a} \right)}^2} - {b^2}}}\)
22.	\({{\bf{e}}^{at}}\cosh \left( {bt} \right)\)	\(\displaystyle \frac{{s - a}}{{{{\left( {s - a} \right)}^2} - {b^2}}}\)
23.	\({t^n}{{\bf{e}}^{at}},\,\,\,\,\,n = 1,2,3, \ldots \)	\(\displaystyle \frac{{n!}}{{{{\left( {s - a} \right)}^{n + 1}}}}\)
24.	\(f\left( {ct} \right)\)	\(\displaystyle \frac{1}{c}F\left( {\frac{s}{c}} \right)\)
25.	\({u_c}\left( t \right) = u\left( {t - c} \right)\)	\(\displaystyle \frac{{{{\bf{e}}^{ - cs}}}}{s}\)
26.	\(\delta \left( {t - c} \right)\)	\({{\bf{e}}^{ - cs}}\)
27.	\({u_c}\left( t \right)f\left( {t - c} \right)\)	\({{\bf{e}}^{ - cs}}F\left( s \right)\)
28.	\({u_c}\left( t \right)g\left( t \right)\)	\({{\bf{e}}^{ - cs}}{\mathcal{L}}\left\{ {g\left( {t + c} \right)} \right\}\)
29.	\({{\bf{e}}^{ct}}f\left( t \right)\)	\(F\left( {s - c} \right)\)
30.	\({t^n}f\left( t \right),\,\,\,\,\,n = 1,2,3, \ldots \)	\({\left( { - 1} \right)^n}{F^{\left( n \right)}}\left( s \right)\)
31.	\(\displaystyle \frac{1}{t}f\left( t \right)\)	\(\int_{{\,s}}^{{\,\infty }}{{F\left( u \right)\,du}}\)
32.	\(\displaystyle \int_{{\,0}}^{{\,t}}{{\,f\left( v \right)\,dv}}\)	\(\displaystyle \frac{{F\left( s \right)}}{s}\)
33.	\(\displaystyle \int_{{\,0}}^{{\,t}}{{f\left( {t - \tau } \right)g\left( \tau \right)\,d\tau }}\)	\(F\left( s \right)G\left( s \right)\)
34.	\(f\left( {t + T} \right) = f\left( t \right)\)	\(\displaystyle \frac{{\displaystyle \int_{{\,0}}^{{\,T}}{{{{\bf{e}}^{ - st}}f\left( t \right)\,dt}}}}{{1 - {{\bf{e}}^{ - sT}}}}\)
35.	\(f'\left( t \right)\)	\(sF\left( s \right) - f\left( 0 \right)\)
36.	\(f''\left( t \right)\)	\({s^2}F\left( s \right) - sf\left( 0 \right) - f'\left( 0 \right)\)
37.	\({f^{\left( n \right)}}\left( t \right)\)	\({s^n}F\left( s \right) - {s^{n - 1}}f\left( 0 \right) - {s^{n - 2}}f'\left( 0 \right) \cdots - s{f^{\left( {n - 2} \right)}}\left( 0 \right) - {f^{\left( {n - 1} \right)}}\left( 0 \right)\)