The Laplace transform is an integral transform perhaps second only to the Fourier
transform in its utility in solving physical problems. The Laplace transform
is particularly useful in solving linear ordinary differential equations such as those arising in the
analysis of electronic circuits.
The (unilateral) Laplace transform (not to be confused
with the Lie derivative, also
commonly denoted ) is defined by
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(1)
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where is defined for (Abramowitz
and Stegun 1972). The unilateral Laplace transform is almost always what is meant
by "the" Laplace transform, although a bilateral Laplace transform is sometimes also defined as
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(Oppenheim et al. 1997). The unilateral Laplace transform is
implemented in Mathematica
as LaplaceTransform[f[t], t, s].
The inverse Laplace transform is known as the Bromwich integral, sometimes known as the Fourier-Mellin integral (see also the related
Duhamel's convolution
principle).
A table of several important one-sided Laplace transforms is given below.
In the above table, is the zeroth-order Bessel function of the first kind, is the
delta function, and is the Heaviside step function.
The Laplace transform has many important properties. The Laplace transform existence theorem states that, if is piecewise continuous on every finite interval in satisfying
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(3)
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for all , then exists
for all . The Laplace transform is also
unique, in the sense that, given two
functions and with the
same transform so that
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then Lerch's theorem guarantees
that the integral
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(5)
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vanishes for all for a null function defined by
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The Laplace transform is linear
since
The Laplace transform of a convolution
is given by
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Now consider differentiation. Let be continuously differentiable times in . If , then
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This can be proved by integration
by parts,
Continuing for higher-order derivatives then gives
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This property can be used to transform differential equations into algebraic equations, a procedure known as the Heaviside
calculus, which can then be inverse transformed to obtain the solution. For example,
applying the Laplace transform to the equation
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(17)
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gives
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which can be rearranged to
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(20)
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If this equation can be inverse Laplace transformed, then the original differential equation is solved.
The Laplace transform satisfied a number of useful properties. Consider exponentiation. If
for (i.e., is the Laplace
transform of ), then
for . This follows from
The Laplace transform also has nice properties when applied to integrals of functions. If is piecewise continuous and ,
then
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(24)
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Abramowitz, M. and Stegun, I. 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.
Graf, U. Applied Laplace Transforms and z-Transforms for Scientists
and Engineers: A Computational Approach using a Mathematica Package.
Basel, Switzerland: Birkhäuser, 2004.
Jaeger, J. C. and Newstead, G. H. An Introduction to the Laplace Transformation with Engineering
Applications. London: Methuen, 1949.
Henrici, P. Applied and Computational Complex Analysis, Vol. 2: Special
Functions, Integral Transforms, Asymptotics, Continued Fractions. New York:
Wiley, pp. 322-350, 1991.
Krantz, S. G. "The Laplace Transform." §15.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser,
pp. 212-214, 1999.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill,
pp. 467-469, 1953.
Oberhettinger, F. Tables of Laplace Transforms. New York: Springer-Verlag,
1973.
Oppenheim, A. V.; Willsky, A. S.; and Nawab, S. H. Signals and Systems, 2nd ed. Upper Saddle River, NJ: Prentice-Hall,
1997.
Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 4: Direct Laplace Transforms.
New York: Gordon and Breach, 1992.
Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 5: Inverse Laplace Transforms.
New York: Gordon and Breach, 1992.
Spiegel, M. R. Theory and Problems of Laplace Transforms. New York: McGraw-Hill,
1965.
Weisstein, E. W. "Books about Laplace Transforms." http://www.ericweisstein.com/encyclopedias/books/LaplaceTransforms.html.
Widder, D. V. The Laplace Transform. Princeton, NJ: Princeton University
Press, 1941.
Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton,
FL: CRC Press, pp. 231 and 543, 1995.
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