“Laplace Transform”的概念、定义、习题解题方法及翻译-数学辞典

单词

Laplace Transform

释义

Laplace Transform

The Laplace transform is an Integral Transform perhaps second only to the Fourier Transform in its utility insolving physical problems. Due to its useful properties, the Laplace transform is particularly useful in solving linearOrdinary Differential Equations such as those arising in the analysis ofelectronic circuits.

The (one-sided) Laplace transform (not to be confused with the Lie Derivative) is defined by

(1)

where

is defined for

. A two-sided Laplace transform is sometimes also defined by

(2)

The Laplace transform existence theorem states that, if

is piecewise Continuous on every finite interval in

satisfying

(3)

for all

, then

exists for all

. The Laplace transform is also Unique, in thesense that, given two functions

and

with the same transform so that

(4)

then Lerch's Theorem guarantees that the integral

(5)

vanishes for all

for a Null Function defined by

(6)

The Laplace transform is Linear since



	(7)

The inverse Laplace transform is given by the Bromwich Integral (see also Duhamel's Convolution Principle).A table of several important Laplace transforms follows.

		Range
1

In the above table, is the zeroth order Bessel Function of the First Kind, is the DeltaFunction, and is the Heaviside Step Function. The Laplace transform has many important properties.

The Laplace transform of a Convolution is given by

(8)

(9)

Now consider Differentiation. Let be continuously differentiable times in . If, then

(10)

This can be proved by Integration by Parts,




			(11)

Continuing for higher order derivatives then gives

(12)

This property can be used to transform differential equations into algebraic equations, a procedure known as theHeaviside Calculus, which can then be inverse transformed to obtain the solution. For example, applying theLaplace transform to the equation

(13)

gives

(14)

(15)

which can be rearranged to

(16)

If this equation can be inverse Laplace transformed, then the original differential equation is solved.

Consider Exponentiation. If for , then for .


			(17)

Consider Integration. If is piecewise continuous and , then

(18)

The inverse transform is known as the Bromwich Integral, or sometimes the Fourier-Mellin Integral.

See also Bromwich Integral, Fourier-Mellin Integral, Fourier Transform, Integral Transform,Laplace-Stieltjes Transform, Operational Mathematics

References

Laplace Transforms

Abramowitz, M. and Stegun, C. A. (Eds.). ``Laplace Transforms.'' Ch. 29 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1019-1030, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 824-863, 1985.

Churchill, R. V. Operational Mathematics. New York: McGraw-Hill, 1958.

Doetsch, G. Introduction to the Theory and Application of the Laplace Transformation. Berlin: Springer-Verlag, 1974.

Franklin, P. An Introduction to Fourier Methods and the Laplace Transformation. New York: Dover, 1958.

Jaeger, J. C. and Newstead, G. H. An Introduction to the Laplace Transformation with Engineering Applications. London: Methuen, 1949.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 467-469, 1953.

Spiegel, M. R. Theory and Problems of Laplace Transforms. New York: McGraw-Hill, 1965.

Widder, D. V. The Laplace Transform. Princeton, NJ: Princeton University Press, 1941.

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