“Laplace transform”的概念、定义、翻译、参考文献-科学参考

Laplace transform

Mathematics

An integral transform ℒ of a function f(x) into another function ℒf or $\bar{f}$ of a different variable p. The Laplace transform $\bar{f} (p)$ of f(x) is defined as

$\bar{f} (p) = \int_{0}^{\infty} f (x) e^{- p x} d x .$

The transform is useful in solving differential equations, as it handles derivatives well; for example, the transform of f′(x) equals $p \bar{f} (p) - f (0)$ . At its simplest, a differential equation in f(x) is transformed into an algebraic equation involving $\bar{f} (p)$ which is solved to find $\bar{f} (p)$ ; by recognizing $\bar{f} (p)$ as the transform of a function, or by applying an inverse transform theorem, the solution f(x) can then be found. For a list of Laplace transforms, see appendix 10. See also Fourier transform.

Chemical Engineering

A method of obtaining a solution to a differential equation where the unknown integration constants are obtained using straightforward algebra. It is commonly (p. 215) used in process control applications since differential equations do not readily enable the relationship between the input and output to be discerned. The Laplace transformation therefore allows a simpler algebraic calculation to be performed. The Laplace transform of a function f(t) is therefore multiplied by e^−st and the product integrated between zero and infinity. It is denoted by $ℒ {f (t)}$ as:
$F (s) = ℒ {f (t)} = \int_{0}^{\infty} e^{- s t} f (t) d t$

where s is a variable whose values are chosen such that the semi-infinite integral converges (i.e. the integration is between 0 and +∞ and is therefore one-sided). For the Laplace transform to exist, the integrand $e^{- s t} f (t)$ must converge to zero as t approaches infinity.

As an example, the Laplace transform of a unit step function is:
$\int_{0}^{\infty} 1 e^{- s t} d t = {[\frac{e^{- s t}}{- s}]}_{0}^{\infty} = \frac{1}{s}$

If F(s) is the Laplace transform of f(t) then f(t) is the inverse Laplace transform of F(s). Thus:
$f (t) = ℒ^{- 1} {F (s)}$

There is no simple definition of the inverse transform and the solution is found in reverse. Tables of Laplace transforms and their inverse transforms are used.

Electronics and Electrical Engineering

A mathematical method of simplifying the transient analysis of a network or circuit. The Laplace transform F(s) of a signal f(t) is given by:
$F (s) = {\int^{}}_{0}^{+ \infty} e^{- s t} f (t) d t$
where s is the complex frequency σ + jω. See also s-domain circuit analysis.

单词	Laplace transform
释义	Laplace transform Mathematics An integral transform ℒ of a function f(x) into another function ℒf or $\bar{f}$ of a different variable p. The Laplace transform $\bar{f} (p)$ of f(x) is defined as $\bar{f} (p) = \int_{0}^{\infty} f (x) e^{- p x} d x .$ The transform is useful in solving differential equations, as it handles derivatives well; for example, the transform of f′(x) equals $p \bar{f} (p) - f (0)$ . At its simplest, a differential equation in f(x) is transformed into an algebraic equation involving $\bar{f} (p)$ which is solved to find $\bar{f} (p)$ ; by recognizing $\bar{f} (p)$ as the transform of a function, or by applying an inverse transform theorem, the solution f(x) can then be found. For a list of Laplace transforms, see appendix 10. See also Fourier transform. Chemical Engineering A method of obtaining a solution to a differential equation where the unknown integration constants are obtained using straightforward algebra. It is commonly (p. 215) used in process control applications since differential equations do not readily enable the relationship between the input and output to be discerned. The Laplace transformation therefore allows a simpler algebraic calculation to be performed. The Laplace transform of a function f(t) is therefore multiplied by e^−st and the product integrated between zero and infinity. It is denoted by $ℒ {f (t)}$ as: $F (s) = ℒ {f (t)} = \int_{0}^{\infty} e^{- s t} f (t) d t$ where s is a variable whose values are chosen such that the semi-infinite integral converges (i.e. the integration is between 0 and +∞ and is therefore one-sided). For the Laplace transform to exist, the integrand $e^{- s t} f (t)$ must converge to zero as t approaches infinity. As an example, the Laplace transform of a unit step function is: $\int_{0}^{\infty} 1 e^{- s t} d t = {[\frac{e^{- s t}}{- s}]}_{0}^{\infty} = \frac{1}{s}$ If F(s) is the Laplace transform of f(t) then f(t) is the inverse Laplace transform of F(s). Thus: $f (t) = ℒ^{- 1} {F (s)}$ There is no simple definition of the inverse transform and the solution is found in reverse. Tables of Laplace transforms and their inverse transforms are used. Electronics and Electrical Engineering A mathematical method of simplifying the transient analysis of a network or circuit. The Laplace transform F(s) of a signal f(t) is given by: $F (s) = {\int^{}}_{0}^{+ \infty} e^{- s t} f (t) d t$ where s is the complex frequency σ + jω. See also s-domain circuit analysis.
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