Mathematical Handbook for Scientists and Engineers

8.1-1. The Laplace transformation (Sec. 8.2-1) associates a unique function F(s) of a complex variable s with each suitable function f(t) of a real variable t. This correspondence is essentially reciprocal one-to-one for most practical purposes (Sec. 8.2-8); corresponding pairs of functions f(t) and F(s) can often be found by reference to tables. The Laplace transformation is defined so that many relations between, and operations on, the functions f(t) correspond to simpler relations between, and operations on, the functions F(s) (Secs. 8.3-1 to 8.3-4). This applies particularly to the solution of differential and integral equations. It is, thus, often useful to transform a given problem involving functions f(t) into an equivalent problem expressed in terms of the associated Laplace transforms F(s) (“operational calculus” based on Laplace transformations or “transformation calculus,” Secs. 9.3-7, 9.4-5, and 10.5-2).

8.2. THE LAPLACE TRANSFORMATION

8.2-1. Definition. The (one-sided) Laplace transformation

associates a unique result or image function F(s) of the complex variable s = σ + iω with every single-valued object or original function f(t) (t real) such that the improper integral (1) exists. F(s) is called the (one-sided) Laplace transform of f(t). The more explicit notation is also used.

8.2-2. Absolute Convergence. The Laplace transform (1) exists for σ ≥ σ_o, and the improper integral converges absolutely and uniformly (Sec. 4.6-2a) to a function F(s) analytic (Sec. 7.3-3) for σ > σ₀ if

exists for σ = σ_o. The greatest lower bound σ_a of the real numbers σ_o for which this is true is called the abscissa of absolute convergence of the Laplace transform .

Although certain theorems relating to Laplace transforms require only the existence (simple convergence) of the transforms, the existence of an abscissa of absolute convergence will be implicitly assumed throughout the following sections.Wherever necessary, it is customary to specify the region of absolute convergence associated with a relation involving Laplace transforms by writing σ > σ_a to the right of the relation in question, as in Eq. (3).

8.2-3. Extension of the Region of Definition. The region of definition of the analytic function

can usually be extended by analytic continuation (Secs. 7.8-1 and 7.8-2) so as to include the entire s plane with the exception of singular points (Sec. 7.6-2) situated to the left of the abscissa of absolute convergence. Such an extension of the region of definition is implied wherever necessary.

8.2-4. Sufficient Conditions for the Existence of the Laplace Transform. The Laplace transform defined by Eq. (1) exists in the sense of absolute convergence (Sec. 8.2-2; see also Sec. 4.9-3)

8.2-5. Inverse Laplace Transformation. The inverse Laplace transform of a (suitable) function F(s) of the complex variable s = σ + iω is a function f(t) whose Laplace transform (1) is F(s). Not every function F(s) has an inverse Laplace transform.

8.2-6. The Inversion Theorem. Given , σ > σ_a, then throughout every open interval where f(t) is of bounded variation [Sec. 4.4-8b; e.g., an open interval where f(t) is bounded and has a finite number of maxima, minima, and discontinuities]

In particular, for every t > 0 where f(t) is continuous

The path of integration in Eq. (4) lies to the right of all singularities of F(s). The inversion integral f₁(t) reduces to if the integral exists; otherwise, f₁(t) is a Cauchy principal value (Sec. 4.6-2b).

8.2-7. Existence of the Inverse Laplace Transform. Note carefully that the existence of the limit (4) for a given function F(s) does not in itself imply that F(s) has an inverse Laplace transform [EXAMPLE: F(s) = e^8²]. The existence of should be checked [e.g., by Eq. (1)] for every application of the inversion theorem. The following theorems state sufficient (not necessary) conditions for the existence of

1. If F(s) is analytic for σ ≥ σ_a and of order < — 1 (Sec. 4.4-3), then exists, is continuous for all t and O(e^σ_at) as t → ∞, and the corresponding abscissa of absolute convergence is σ_a. Note that, under these conditions, Eq. (4b) holds, and

2. Given F(s) = φ[F₁(s), F₂(s), . . . , F_n(s)] such that φ(z₁, z₂, . . . , z_n) is equal to zero and analytic with respect to each z_k for z₁ = z₂ … = z_n = 0, and then exists, and the corresponding Laplace transformation possesses an abscissa of absolute convergence.

8.2-8. Uniqueness of the Laplace Transform and Its Inverse. The Laplace transform (1) is unique for each function f(t) having such a transform. Conversely, two functions f₁(t) and f₂(t) possessing identical Laplace transforms are identical for all t > 0, except possibly on a set of measure zero (Sec. 4.6-14); f₁(t) = f₂(t) for all t > 0 where both functions are continuous (Lerch’s Theorem). Thus, f(t) is uniquely defined by its Laplace transform for almost all t > 0 (Sec. 4.6-14b); a given function F(s) cannot have more than one inverse Laplace transform continuous for all t > 0.

Different discontinuous functions may have the same Laplace transform. In particular, the generalized unit step function (see also Sec. 21.9-1) defined by f(t) = 0 for t <0, f(t) = 1 for t > 0 has the Laplace transform 1/s regardless of the value assigned to f(t) for t = 0.

8.3. CORRESPONDENCE BETWEEN OPERATIONS ON OBJECT AND RESULT FUNCTIONS

8.3-1.Table of Corresponding Operations. Table 8.3-1 lists a number of theorems each establishing a correspondence between an operation on a function f₁(t)and an operation on its Laplace transform F₁(s), and vice versa. These theorems are the basis of Laplace-transform techniques for the simplified representation of operations (operational calculus based on the use of Laplace transforms).

8.3-2. Laplace Transforms of Periodic Object Functions and Amplitude-modulated Sinusoids. (a) If f(t) is periodic with period T (Sec. 4.2-2b), and exists, then

NOTE: The integral on the right is an integral function (Sec. 7.6-5), so that has no singularities for finite s except possibly for simple poles on the imaginary axis.

(b) If f(t) is antiperiodic (Sec. 4.2-2b) with period T, and exists, then

(c) If φ₁(t) is the result of ideal half-wave rectification of the antiperiodic function f(t) where both f(t) and φ₁(t) are positive for 0 < t < T/2, then

(d) if φ₂ is the result of ideal full-wave rectification of the antiperiodic function f(t) where both f(t) and φ₂(t) are positive for 0 < t < T/2, then

(e) Transform of an Amplitude-modulated Sinusoid. If F(s) is the Laplace transform of f(t), then

Table 8.3-1. Theorems Relating Corresponding Operations on Object and Result Functions

The following theorems are valid whenever the Laplace transforms F(s) = £[f(t)] in question exist in the sense of absolute convergence; limits are assumed to exist (see also Secs. 8.3-2 and 8.7-3)

* The abscissa of absolute convergence for £[f^(r)(t)] is 0 or σ_a, whichever is greater.

† The existence of f₁ * f₂ is assumed; absolute convergence of £[f₁(t)] and £[f₂(t) is a sufficient condition for the absolute convergence of £[f₁ * f₂]. See also Sec. 8.3-3.

8.3-3. Transform of a Product (Convolution Theorem). Given

8.3-4. Limit Theorems. If F(s) is the Laplace transform of f(t) and exists, then

if the limit on the left exists. If, in addition to the first two conditions, sF(s) is analytic for σ ≥ 0, then

(see also Table 8.3-1,2).

8.4. TABLES OF LAPLACE-TRANSFORM PAIRS AND COMPUTATION OF INVERSE LAPLACE TRANSFORMS

8.4-1. Tables of Laplace-transform Pairs. A number of Laplace-transform pairs are tabulated in Appendix D for reference. In particular, Table D-6 lists a number of Laplace-transform pairs with rational algebraic result functions. Appendix D also shows how tables of Fourier-transform pairs may be used to obtain certain Laplace-transform pairs, and vice versa.

8.4-2. Computation of Inverse Laplace Transforms. Sections 8.4-3 to 8.4-9 describe various procedures for finding the object function f(t) corresponding to a result function F(s) obtained in the course of a problem solution (see also Secs. 9.3-7, 9.4-5, and 10.5-2).

NOTE: Unless the existence of is definitely known, any results obtained through the use of the inversion integral (8.2-4) should be checked by means of Eq. (8.2-1). Particular caution is recommended in connection with the use of series expansions for seemingly straightforward asymptotic or even convergent expansions may not be valid unless F(s) and f(t) satisfy certain restrictive conditions. A number of sufficient (but not necessary) conditions for the validity of series expansions of are presented in Secs. 8.4-6 to 8.4-9. In many problems, a function f(t) “suspected” to be can be tested, e.g., by resubstitution into a differential equation, so that heuristic methods for finding may be quite useful.

8.4-3. Use of Contour Integration. Values of the contour integral (8.2-4) may frequently be obtained with the aid of the residue theorem (Secs. 7.7-1 to 7.7-3) and Jordan’s lemma (Sec. 7.7-3b). If F(s) is a multiple-valued function, the contours used must not cross any branch cuts of F(s) (Sec. 7.4-2; see Refs. 8.5 and 8.10).

8.4-4. Inverse Laplace Transforms of Rational Algebraic Functions : Heaviside Expansion. (a) If F(s) is a rational algebraic function expressible as the ratio of two polynomials in s,

such that the degree of the polynomial D(s) is higher than that of the polynomial D₁(s), then equals the sum of the residues ( Sec. 7.7-1) of F(s)e^8t at all the singular points (poles) of F(s). To compute the inverse Laplace transform , first find the roots s_k of D(s) = 0 [which determine the poles of F(s)] by one of the methods described in Secs. 1.8-1 to 1.8-6 or 20.2-2 and 20.2-3. Then

(see also Table D-6).

(b) It is sometimes convenient to obtain an inverse Laplace transform involving a multiple root of D(s) =0 directly by a limiting process leading to the coincidence of distinct simple roots (see also Table 8.3-1,7). EXAMPLE:

and, more generally,

(c)Complex Roots. In Eq. (2), pairs of terms corresponding to complex-conjugate roots s = a ± iω₁ may be combined as follows (see also Sec. 8.4-5):

A, B, R, α and α′ are real if the coefficients in D₁(s) and D(s) are real; is then a real function of t.

(d) If one of the roots s_k of D(s) = 0 should also be a root of D₁(s) = 0, then one or more terms of the expansion (2) will vanish. In general, such common roots of D(s) = 0 and D₁(s) = 0 can be “factored out” and canceled in Eq. (1).

8.4-5. Inverse Laplace Transforms of Rational Algebraic Functions: Expansion in Partial Fractions. Instead of applying the Heaviside expansion (2) directly, one may expand as a sum of partial fractions by one of the methods described in Sec. 1.7-4. If D(s) and D₁(s) have no common zeros, each real root s_{_k} = a of D(s) = 0 will give rise to m_{_k} partial fractions of the form

where m_k is the order of the root s_k = a. Each pair of complex-conjugate roots S_k = a ± iω₁ will give rise to m_k partial fractions of the form

where m_k is the order of the roots S_k = a ± iω₁. is then obtained as the sum of the inverse Laplace transforms of such terms (Table D-6). The method of Sec. 8.4-4b may be useful.

8.4-6. Expansions in Series. If the form of F(s) is complicated, or if F(s) is given only implicitly, e.g., as the solution of a differential equation in s (Sec. 9.4-5), it is sometimes possible to obtain by expanding F(s) into a convergent series and taking the inverse Laplace transform of the latter term by term. Such a procedure may also be useful for approximating . Frequently, series-expansion methods are justified by the following theorem. Let

and let

converge. Then the series converges uniformly to a function F(s) for σ > σ_a; the series converges absolutely to a function f(t) for almost all t (Sec.4.6-14b), and

8.4-7. Expansions in Terms of Powers of t. Series expansions of F(s) in descending powers of s,

may frequently be obtained as Laurent expansions (Sec. 7.5-3) about s = 0 or, in the case of rational algebraic functions of the type specified in Sec. 8.4-4, simply by long division (Sec. 1.7-2). If the conditions of Sec. 8.4-6 are satisfied, then, for almost all t in (0, t₁),

which may furnish (at least) useful approximations to for t < t₁.

converges, then

In particular, if the series on the left converges,

8.4-8. Expansions in Terms of Laguerre Polynomials of t. Every function

analytic at s = ∞ may be expanded into an absolutely convergent series in powers of

for σ > σ_a, corresponding to a Taylor series convergent for |z| <1. In particular, for σ_a = 0, F(s) may be expressed in the form

and, if the conditions of Sec. 8.4-6 are satisfied, then, for almost all t > 0,

where the Lk(t) are the Laguerre polynomials defined in Sec. 21.7-1.

8.4-9. Expansions in Asymptotic Series. Valid approximations to may often be obtained in terms of asymptotic series (Sec. 4.8-6b). The following type of asymptotic expansion is of importance in connection with the solution of certain partial differential equations (Sec. 10.5-2).

can be expanded into a convergent series of the form

in a neighborhood of s = 0, then f(t) may be represented by the asymptotic series

provided that either of the following sets of conditions is satisfied (see also Sec. 8.5-1 and Ref.8.5):

1. f(t) is differentiable for t > 0, continuous for t = 0, and exists; and there exists a (finite) real positive number K such that where S_j—1(t) is the (j — l)^st partial sum of the series in Eq. (14). 2.

2. f(t) is continuous for t > t1 > 0, and σ_a ≤ 0; F(s) is analytic for σ ≥ 0 except for s = 0. There exists σ₁ > 0 such that for σ₁ ≥ σ ≥ 0

and converges uniformly for t > t₁ > 0.

These rather restrictive conditions provide a rigorous basis for a number of asymptotic expansions of originally derived with the aid of the old Heaviside operational calculus (Refs. 8.3 and 8.10).

8.5. “FORMAL” LAPLACE TRANSFORMATION OF IMPULSE-FUNCTION TERMS

8.5-1. Impulse-function Transforms. (a) In Eq. (8.4-13) and in similar series expansions, note that the inverse Laplace transforms of individual terms like a, as, as^3/2, . . . do not, strictly speaking, exist, since these functions do not tend to zero as s → ∞. In most applications, terms of this nature will appear under summation and integral signs in such a manner that the series or integral does have an inverse Laplace transform.

(b) If one applies the Laplace transformation (8.2-1) to the “definitions” of the impulse functions δ(t) and δ₊(t) and their “derivatives” (Secs. 9.2 and 21.9-6), one obtains the formal results

and, if f(t) is continuous for t = a ≥ 0,

Eqs. (1) and (2) are useful in many applications, but one must not forget that such relations have no strict mathematical meaning. New results suggested by Eqs. (1) and (2) must always be verified by mathematically legitimate means (see also Secs. 8.2-7 and 21.9-2a).

8.6. SOME OTHER INTEGRAL TRANSFORMATIONS

8.6-1. The Laplace transformation (8.2-1) is a functional transformation associating “points” F(s) in a result space with “points” f(t) in an object space (see also Secs. 12.1-4 and 15.2-7). Table 8.6-1 and Secs. 8.6-2 to 8.6-4 introduce a number of other functional transformations (see also Sec. 4.11-5, Appendix D, and Refs. 8.8 and 8.10).

* The asymmetrical impulse function δ₊(t) is more suitable for use in connection with the one-sided Laplace transformation than the symmetrical impulse function δ(t).

8.6-2. The Two-sided (Bilateral) Laplace Transformation. (a) The two-sided (bilateral) Laplace transformation

is an attractive generalization of the Laplace transformation applicable, like the Fourier transformation (Secs. 4.11-3 to 4.11-7), to problems where values of f(t) for t < 0 are of importance. converges absolutely if and only if both and converge absolutely, so that the region of absolute convergence, if any, will be a strip of the s plane determined by two abscissas of absolute convergence. Many properties of the two-sided Laplace transformation are simply derived from corresponding properties of the one-sided Laplace transformation by reference to Eq. (1) (see also Refs. 8.5 and 8.17). In particular, note

(b) The values of the inverse transform are not restricted to zero for t < 0, so that exists for a larger class of functions F(s) than does . Given

then, for any value of t having a neighborhood where f(t) is of bounded variation,

(c) Note also the convolution theorem

with

assuming absolute convergence (see also Tables 4.11-1 and 8.3-1).

8.6-3. The Stieltjes-integral Form of the Laplace Transformation. The Stieltjes-integral form of the Laplace transformation

Table 8.6-1. Some Linear Integral Transformations Related to the Laplace Transformation

Each transform is denoted by F(s) (see also Secs. 4.11-3 to 4.11-5, 8.6-1 to 8.6-5, 10.5-1, 15.2-7, 15.3-1, and 18.3-8)

* Whenever the transforms in question exist.

permits a more general formulation of many theorems than the ordinary Laplace transformation (see also Sec. 4.6-17 and Refs. 8.5 and 8.18). The other functional transformations listed in Table 8.6-1 may be similarly written in terms of Stieltjes integrals.

Note that F(s) ≡ se^—b8 can be represented in the form (5) without the use of impulse functions (Sec. 8.5-1). The s-multiplied Laplace transform (Table 8.6-1,1) is sometimes employed for similar reasons.

8.6-4. Hankel Transforms and Fourier-Bessel Transforms. (a) Definition and Inversion Theorem. The integral transform

where f(t) is a real function, and J_m(z) is the m^th-order Bessel function (Sec. 21.8-1), exists in the sense of absolute convergence whenever exists. If, in addition, f(t) is of bounded variation in a neighborhood of t, one has the inversion formula

which determines the inverse transform uniquely wherever it is continuous.

(b) Properties of Hankel Transforms. Note the following relations:

(c) Fourier-Bessel Transforms (see also Secs. 21.8-1 and 21.8-2). The following integral-transform pairs are related to the Hankel-transform pair (6):

Both integral-transform pairs are referred to as Fourier-Bessel transform pairs; for m = 0, Eq. (13) reduces to a Fourier sine-transform pair.

8.7. FINITE INTEGRAL TRANSFORMS, GENERATING FUNCTIONS, AND z TRANSFORMS

8.7-1. Series as Functional Transforms. Finite Fourier and Hankel Transforms. A finite or convergent series

represents a functional transformation of the function (sequence, Sec. 4.2-1) f_k ≡ f(k) defined on the discrete set of integers k = 0, 1, 2, . . . . Note that for suitable f_k and Ψ(x,k) such a series can be written as an integral transform in terms of a Stieltjes integral (Sec. 4.6-17):

The series (1) and formulas like (4.10-5), (4.11-6), (7.5-4), and (7.5-7), which relate series coefficients f_k to a function Φ(x), constitute corresponding inverse functional transformations. Table 8.7-1 lists relations for generalized Fourier-series coefficients regarded as integral transforms with finite integration intervals (finite integral transforms), and for the analogous finite Hankel transforms. In each case, the transformation of a relevant second-order differential equation is illustrated for use in connection with the solution of boundary-value problems (Refs. 8.15 and 8.16; see also Secs. 10.4-9, 15.2-4, and 21.8-4c).

8.7-2. Generating Functions. If the functional transformation (1) takes the form of a finite or convergent power series

γ(s) is called a generating function for the sequence of coefficients f₀, f₁ f₂ ... , while

Table 8.7-1. Some Finite Integral Transforms (see also Secs. 10.4-2c and 15.4-12). x is a real variable, Φ'(x) ≡ dØ/dx, Φ"(x) ≡ d²Φ/dx², and a > 0.

Finite Transforms Useful with Boundary Conditions of the Form Φ = 0 or Φ' = 0

In Tables 8.7-1a and b, note that J'₀(λ_ka) ≡ -j₁(λ_ka).

is called an exponential generating function for the f_{_k}. Applications and properties of generating functions are further discussed in Sec. 18.3-8 and Appendix C.

EXAMPLE: Fibonacci numbers f₀, f₁ f₂, . . . are defined by

Their generating function is

8.7-3. z Transforms. Definition and Inversion Integral. The z transform of a suitable sequence f₀, f₁, f₂ . . . is defined as

where z is a complex variable, and the series converges absolutely outside of a circle of absolute convergence of radius r_a depending on the given sequence; analytic continuation in the manner of Sec. 8-2.3 can extend the definition. The corresponding inversion integral

where C is a closed contour enclosing all singularities of Fz(z), gives the inverse transform Z^–1[F_z(z)] = f_k for suitable f_k (Ref. 8.11). Inversion can then utilize the residue calculus in the manner of Sec. 8.4-3, especially if F_z(z) is a rational function expandable in partial fractions. Inversion is even simpler if F_z(z) can be expanded directly in terms of powers of 1/z. Note that the inverse transform must be unique wherever the series (6) converges absolutely (Sec. 4.10-2c).

Table 8.7-2 summarizes the most important properties of z transforms. Their application to the solution of difference equations and sampled-data systems is treated in Sec. 20.4-6, where the relation of z transforms to jump-function Laplace tranforms is also discussed. Table 20.4-1 lists a number of z-transform pairs.

The z transform is related to the Mellin transform of Table 8.6-1. Note the analogy between power series and Mellin transforms and between Dirichlet series and Laplace transforms.

8.8. RELATED TOPICS, REFERENCES, AND BIBLIOGRAPHY

8.8-1. Related Topics. The following topics related to the study of the Laplace transformation and other functional transformations are

Table 8.7-2. Corresponding Opeations for z Transforms

The following theorems are valid whenever the z-transform series in question converge absolutely; limits are assumed to exist (see also Table 8.3-1), and f_-1 = f_-2 = . . . = g_-1 = g_-2 = . . . = 0

* (z-1)Fz(z) is assumed to be analytic for |z| ≥ 1.

treated in other chapters of this handbook:

Expansion in partial fractions Chap. 1

Limits, integration, and improper integrals Chap. 4

Convergent and asymptotic series Chap. 4

Functions of a complex variable, contour integration Chap. 7

Transformations Chaps. 12, 14, 15

Fourier transforms Chap. 4

Applications of generating functions Chap. 18, Appendix C

z transforms Chap. 20

Tables of Fourier, Hankel, and Laplace-transform pairs Appendix D

Applications of the Laplace transformation are discussed in other chapters of this handbook as follows:

Ordinary differential equations Chaps. 9, 13

Partial differential equations Chap. 10

Integral equations Chap. 15

8.8-2. References and Bibliography.

      8.1. Campbell, G. A., and R. M. Foster: Fourier Integrals for Practical Applications, Van Nostrand, Princeton, N.J., 1958.

      8.2. Churchill, R. V.: Operational Mathematics, 2d ed., McGraw-Hill, New York, 1958.

      8.3. Ditkin, V. A., and A. P. Prudnikov: Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965.

      8.4. Doetsch, G.: Anleitung zum Praktischen Gebranch der Laplace-Transformation, Oldenburg, Munich, 1956.

      8.5. : Handbuch der Laplace-Transformation, 3 vols., Birkhauser, Basel, 1950.

      8.6. and D. Voelker: Die Zweidimensionale Laplace-Transformation, Birkhauser, Basel, 1950.

      8.7. , H. Kneiss, and D. Voelker: Tabellen zur Laplace-Transformation und Anleitung zum Gebrauch, Springer, Berlin, 1947.

      8.8. Erdelyi, A., et al.: Tables of Integral Transforms (Bateman Project), 2 vols., McGraw-Hill, New York, 1954.

      8.9. Jury, E. I.: Theory and Application of the z-transform Method, Wiley, New York, 1964.

      8.10. McLachlan, N. W.: Modern Operational Calculus, Macmillan, New York, 1948.

      8.11. Miles, J. W.: Integral Transforms, in E. F. Beckenbach, Modern Mathematics for the Engineer, 2d series, McGraw-Hill, New York, 1961.

      8.12. Nixon, F. E.: Handbook of Laplace Transformation, 2d ed., Prentice-Hall, Englewood Cliffs, N.J., 1965.

      8.13. Papoulis, A.: The Fourier Integral and Its Applications, McGraw-Hill, New York, 1962.

      8.14. Scott, E. J.: Transform Calculus, Harper, New York, 1955.

      8.15. Sneddon, I. N.: Fourier Transforms, McGraw-Hill, New York, 1951.

      8.16. Tranter, C. J.: Integral Transforms in Mathematical Physics, 2d ed., Wiley, New York, 1956.

      8.17. Van Der Pol, B., and H. Bremmer: Operational Calculus Based on the Two-sided Laplace Integral, Cambridge University Press, London, 1950.

      8.18. Widder, D. V.: The Laplace Transform, Princeton University Press, Princeton, N.J., 1941.