18.1: A.1- Table of Laplace Transform Pairs
- Page ID
- 7740
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General conventions:
- time \(t\) is a real number, \(t \geq 0\);
- Laplace variable \(s\) is a complex number with dimension of time-1;
- \(n\) and \(k\) are positive, real integers;
- \(p\) and \(\sigma\) are finite constants, with dimension of time-1;
- \(t_{s}\) is a real, finite constant, with dimension of time;
- \(\omega\) is a positive, real, finite constant, with dimension of time-1.
| Function of time \(t\), inverse Laplace transform | Function of Laplace variable \(s\), forward Laplace transform | Source and notes |
|---|---|---|
| \(f(t)=L^{-1}[F(s)]\) | \(F(s)=L[f(t)]\) | General notation, Section 2.2 |
| \(f(t)\) | \(\int_{t=0}^{t=\infty} e^{-s t} f(t) d t, \text { or } \int_{t=0^{-}}^{t=\infty} e^{-s t} f(t) d t\) | Definitions: Equation 2.2.5, or Equation 8.4.5 if \(f (t)\) involves \(\delta(t)\) |
| \(\dot{f}(t)\) | \(s F(s)-f\left(0^{-}\right)\) | Equations 2.2.9 and 8.6.3; \(t=0^{-}\) for ICs precedes all inputs |
| \(\dot{f}(t)\) | \(s^{2} F(s)-s f\left(0^{-}\right)-\dot{f}\left(0^{-}\right)\) | Equation 2.2.11, Problem 2.4 |
| \(\frac{d^{n}}{d t^{n}} f(t)\) | \(\begin{array}{l} s^{n} F(s)-s^{n-1} f\left(0^{-}\right)-s^{n-2} \dot{f}\left(0^{-}\right)- \\ \cdots-f\left(0^{-}\right) \end{array}\) |
Equations 2.2.10 and 8.6.4 Ogata, 1998, pp. 25- 26 |
| \(\int_{\tau=-\infty}^{\tau=t \geq 0} f(\tau) d \tau\) | \(\frac{1}{s} F(s)+\frac{1}{s} \int_{\tau=-\infty}^{\tau=0} f(\tau) d \tau\) | Equation 2.4.9, derivation in Section 18.3; note that \(-\infty<t<+\infty\) |
| \(\int_{\tau=0}^{\tau=t \geq 0} f(\tau) d \tau\) | \(\frac{1}{s} F(s)\) | Equation 2.4.10, derivation in Section 18.3 |
| \(\begin{aligned} C I(t) &=\int_{\tau=0}^{\tau=t} f_{1}(\tau) f_{2}(t-\tau) d \tau \\ &=\int_{\tau=0}^{\tau=t} f_{1}(t-\tau) f_{2}(\tau) d \tau \end{aligned}\) |
\(F_{1}(s) \times F_{2}(s)\) | Equation 6.1.4, convolution integral, derived in Section 18.5 |
| \(e^{\sigma t} f(t)\) | \(F(s-\sigma)\) | Equation 9.3.6; \(\sigma\) is any constant |
| \(e^{p t}\) | \(\frac{1}{s-p}\) | Equation 2.2.6; \(p\) is any constant |
| \(\frac{1}{p}\left(e^{p t}-1\right)\) | \(\frac{1}{s} \frac{1}{s-p}\) | Eq. (A-15); \(p\) is any constant |
| \(f(t)=L^{-1}[F(s)]\) | \(F(s)=L[f(t)]\) | General notation, Section 2.2 |
| \(f(t)\) | \(\int_{t=0}^{t=\infty} e^{-s t} f(t) d t, \text { or } \int_{t=0^{-}}^{t=\infty} e^{-s t} f(t) d t\) | Definitions: Equation 2.2.5, or Equation 8.4.5 if \(f(t)\) involves \(\delta (t)\) |
| \(\delta(t) \equiv \delta(t-0)\) | \(1\) | Equation 8.4.6; Dirac delta function of Equation 8.4.1 |
| \(H\left(t-t_{s}\right)\) | \(\frac{e^{-s t_{s}}}{s}\) | Equation 2.4.4; general unit-step defined in Equation 2.4.2 |
| \(H(t)\) | \(\frac{1}{s}\) | Equation 2.4.5; Heaviside unit-step function defined in Equation 2.4.1 |
| \(f\left(t-t_{s}\right) H\left(t-t_{s}\right)\) | \(e^{-s t_{s}} F(s)\) | Ogata, 1998, p. 18; translated function of time |
| \(t\) | \(\frac{1}{s^{2}}\) | Equation (A-17) |
| \(t e^{\sigma t}\) | \(\frac{1}{(s-\sigma)^{2}}\) | Equations (A-17) and 9.3.6; \(\sigma\) is any constant |
| \(\sin \omega t\) | \(\frac{\omega}{s^{2}+\omega^{2}}\) | Equation 2.4.7; \(\omega\) is a positive real constant |
| \(\cos \omega t\) | \(\frac{s}{s^{2}+\omega^{2}}\) | Equation 2.4.8; \(\omega\) is a positive real constant |
| \(\frac{1}{\omega^{2}}(1-\cos \omega t)\) | \(\frac{1}{s\left(s^{2}+\omega^{2}\right)}\) | Equation (A-16); \(\omega\) is a positive real constant |
| \(\frac{\omega_{n}^{2}}{\omega_{d}} e^{-\zeta \omega_{n} t} \sin \omega_{d} t\) | \(\frac{\omega_{n}^{2}}{\left(s+\zeta \omega_{n}\right)^{2}+\omega_{d}^{2}}\) | Problem 9.12; \(|\zeta|<1\), \(\omega_{d}^{2}=\omega_{n}^{2}\left(1-\zeta^{2}\right)>0\) |
| \(\begin{array}{l} 1-e^{-\zeta \omega_{n} t} \times \\ \left(\cos \omega_{d} t+\frac{\zeta \omega_{n}}{\omega_{d}} \sin \omega_{d} t\right) \end{array}\) |
\(\frac{\omega_{n}^{2}}{s\left[\left(s+\zeta \omega_{n}\right)^{2}+\omega_{d}^{2}\right]}\) | Problem 9.12; \(|\zeta|<1\), \(\omega_{d}^{2}=\omega_{n}^{2}\left(1-\zeta^{2}\right)>0\) |
| \(\begin{array}{l} e^{-\zeta \omega_{n} t} \times \\ \left(\omega_{d} \cos \omega_{d} t-\zeta \omega_{n} \sin \omega_{d} t\right) \end{array}\) |
\(\frac{s \omega_{d}}{\left(s+\zeta \omega_{n}\right)^{2}+\omega_{d}^{2}}\) | Problem 9.15; \(|\zeta|<1\), \(\omega_{d}^{2}=\omega_{n}^{2}\left(1-\zeta^{2}\right)>0\) |
| \(2|C| e^{\sigma t} \cos (\omega t+\angle C)\) | \(\frac{C}{s-p}+\frac{\bar{C}}{s-\bar{p}}\) | Equation 16.1.12; \(C\) is complex, \(p=\sigma+j \omega\), \(\sigma\) and \(\omega\) are real, \(\omega>0\) |
| \(\sum_{k=1}^{n} \frac{N u m\left(p_{k}\right)}{\operatorname{Den}^{\prime}\left(p_{k}\right)} e^{p_{k} t}\) | \(\frac{\text {Num}(s)}{\operatorname{Den}(s)}\) | Eq. (A-7), Section 18.2 for definitions, derivation, and restrictions |
| \(f(t)=L^{-1}[F(s)]\) | \(F(s)=L[f(t)]\) | General notation, Section 2.2 |
| \(f(t)\) | \(\int_{t=0}^{t=\infty} e^{-s t} f(t) d t, \text { or } \int_{t=0^{-}}^{t=\infty} e^{-s t} f(t) d t\) | Definitions: Equation 2.2.5, or Equation 8.4.5 if \(f(t)\) involves \(\delta (t)\) |
| \(\lim _{t \rightarrow 0^{+} \text {from } t>0} f(t) \equiv f\left(0^{+}\right)\) | \(\lim _{s \rightarrow \infty}[s F(s)]\) | Initial-value theorem: Equation 8.6.1 and Equation 8.6.6 in Section 8.6 |
| \(\lim _{t \rightarrow \infty} f(t)\) | \(\lim _{s \rightarrow 0}[s F(s)]\) | Final-value theorem: Equation 15.2.13 and Section 15.3, valid only if \(\lim _{t \rightarrow \infty} f(t)\) is a finite, constant value |
It is worthy of note that MATLAB’s symbolic software, which is introduced in homework Problem 1.6, can sometimes be helpful for finding forward and inverse Laplace transforms by applying, respectively, the laplace and ilaplace commands. The following are two relatively simple examples that do not appear explicitly in the table of transform pairs. In these examples, MATLAB finds the forward transform \(L[\sinh (a t)]= a /\left(s^{2}-a^{2}\right)\), and the inverse transform \(L^{-1}\left[(s+a) /\left[(s+a)^{2}+\omega^{2}\right]\right]=e^{-a t} \cos \omega t\).
>> syms s t a w
>> ft=sinh(a*t)
ft =
sinh(a*t)
>> Lft=laplace(ft,t,s)
Lft =
a/(s^2-a^2)
>> pretty(Lft)
a
-------
2 2
s - a
>> Fs=(s+a)/((s+a)^2+w^2)
Fs =
(s+a)/((s+a)^2+w^2)
>> fFs=ilaplace(Fs,s,t)
fFs =
exp(-a*t)*cos(w*t)
For functions of time \(t\) or functions of Laplace variable s that are more complicated than those in the examples above, MATLAB might produce solutions that are correct, but are expressed in an unfamiliar form, or in a long and unwieldy form that must be simplified by human touch in order to become useful.
1Much more extensive tables of Laplace transform pairs are available in many references, for example, Cannon, 1967, Appendix J.