Laplace Transform

The Laplace transform can be used to solve differential equations. Besides being a different and efficient alternative to variation of parameters and undetermined coefficients, the Laplace method is particularly advantageous for input terms that are piecewise-defined, periodic or impulsive. The direct Laplace transform or the Laplace integral of a function f (t) defined for 0 ≤ t < ∞ is the ordinary calculus integration problem
∞ 0

f (t)e−st dt,

succinctly denoted L(f (t)) in science and engineering literature. The L–notation recognizes that integration always proceeds over t = 0 to t = ∞ and that the integral involves an integrator e−st dt instead of the usual dt. These minor differences distinguish Laplace integrals from the ordinary integrals found on the inside covers of calculus texts.

7.1 Introduction to the Laplace Method
The foundation of Laplace theory is Lerch’s cancellation law
∞ −st dt 0 y(t)e

=

∞ −st dt 0 f (t)e

(1) L(y(t) = L(f (t))

implies or implies

y(t) = f (t), y(t) = f (t).

In differential equation applications, y(t) is the sought-after unknown while f (t) is an explicit expression taken from integral tables. Below, we illustrate Laplace’s method by solving the initial value problem y = −1, y(0) = 0. The method obtains a relation L(y(t)) = L(−t), whence Lerch’s cancellation law implies the solution is y(t) = −t. The Laplace method is advertised as a table lookup method, in which the solution y(t) to a differential equation is found by looking up the answer in a special integral table.

7.1 Introduction to the Laplace Method

247

Laplace Integral. The integral

is called the Laplace N integral of the function g(t). It is defined by limN →∞ 0 g(t)e−st dt and depends on variable s. The ideas will be illustrated for g(t) = 1, g(t) = t and g(t) = t2 , producing the integral formulas in Table 1.
∞ −st dt 0 (1)e

∞ −st dt 0 g(t)e

= −(1/s)e−st = 1/s = = = =

t=∞ t=0

Laplace integral of g(t) = 1. Assumed s > 0. Laplace integral of g(t) = t. Use d ds F (t, s)dt

∞ −st dt 0 (t)e

∞ d −st )dt 0 − ds (e d ∞ − ds 0 (1)e−st dt d − ds (1/s) 1/s2 ∞ d −st )dt 0 − ds (te d ∞ − ds 0 (t)e−st dt d − ds (1/s2 ) 2/s3

=

d ds

F (t, s)dt.

Use L(1) = 1/s. Differentiate. Laplace integral of g(t) = t2 . Use L(t) = 1/s2 .

∞ 2 −st dt 0 (t )e

= = = =

Table 1. The Laplace integral

∞ 0

g(t)e−st dt for g(t) = 1, t and t2 .

∞ −st 0 (1)e

dt =

1 s

∞ −st 0 (t)e

dt =

1 s2 n! s1+n

∞ 2 −st dt 0 (t )e

=

2 s3

In summary, L(tn ) =

An Illustration. The ideas of the Laplace method will be illustrated for the solution y(t) = −t of the problem y = −1, y(0) = 0. The method, entirely different from variation of parameters or undetermined coefficients, uses basic calculus and college algebra; see Table 2.
Table 2. Laplace method details for the illustration y = −1, y(0) = 0.

y (t)e−st = −e−st
∞ −st dt = ∞ −e−st dt 0 y (t)e 0 ∞ y (t)e−st dt = −1/s 0 ∞ s 0 y(t)e−st dt − y(0) = −1/s ∞ −st dt = −1/s2 0 y(t)e ∞ −st dt = ∞ (−t)e−st dt 0 y(t)e 0

Multiply y = −1 by e−st . Integrate t = 0 to t = ∞. Use Table 1. Integrate by parts on the left. Use y(0) = 0 and divide. Use Table 1. Apply Lerch’s cancellation law.

y(t) = −t

248

Laplace Transform

In Lerch’s law, the formal rule of erasing the integral signs is valid provided the integrals are equal for large s and certain conditions hold on y and f – see Theorem 2. The illustration in Table 2 shows that Laplace theory requires an in-depth study of a special integral table, a table which is a true extension of the usual table found on the inside covers of calculus books. Some entries for the special integral table appear in Table 1 and also in section 7.2, Table 4. The L-notation for the direct Laplace transform produces briefer details, as witnessed by the translation of Table 2 into Table 3 below. The reader is advised to move from Laplace integral notation to the L–notation as soon as possible, in order to clarify the ideas of the transform method.
Table 3. Laplace method L-notation details for y = −1, y(0) = 0 translated from Table 2.

L(y (t)) = L(−1) L(y (t)) = −1/s sL(y(t)) − y(0) = −1/s L(y(t)) = y(t) = −t −1/s2 L(y(t)) = L(−t)

Apply L across y = −1, or multiply y = −1 by e−st , integrate t = 0 to t = ∞. Use Table 1. Integrate by parts on the left. Use y(0) = 0 and divide. Apply Table 1. Invoke Lerch’s cancellation law.

Some Transform Rules. The formal properties of calculus integrals plus the integration by parts formula used in Tables 2 and 3 leads to these rules for the Laplace transform: L(f (t) + g(t)) = L(f (t)) + L(g(t)) L(cf (t)) = cL(f (t)) L(y (t)) = sL(y(t)) − y(0) The integral of a sum is the sum of the integrals. Constants c pass through the integral sign. The t-derivative rule, or integration by parts. See Theorem 3. Lerch’s cancellation law. See Theorem 2.

L(y(t)) = L(f (t)) implies y(t) = f (t)

1 Example (Laplace method) Solve by Laplace’s method the initial value problem y = 5 − 2t, y(0) = 1. Solution: Laplace’s method is outlined in Tables 2 and 3. The L-notation of
Table 3 will be used to find the solution y(t) = 1 + 5t − t2 .

7.1 Introduction to the Laplace Method
L(y (t)) = L(5 − 2t) 2 5 L(y (t)) = − 2 s s 5 2 sL(y(t)) − y(0) = − 2 s s 1 5 2 L(y(t)) = + 2 − 3 s s s L(y(t)) = L(1) + 5L(t) − L(t2 ) y(t) = 1 + 5t − t = L(1 + 5t − t )
2 2

249

Apply L across y = 5 − 2t. Use Table 1. Apply the t-derivative rule, page 248. Use y(0) = 1 and divide. Apply Table 1, backwards. Linearity, page 248. Invoke Lerch’s cancellation law.

2 Example (Laplace method) Solve by Laplace’s method the initial value problem y = 10, y(0) = y (0) = 0. Solution: The L-notation of Table 3 will be used to find the solution y(t) = 5t2 . sL(y (t)) − y (0) = L(10) s2 L(y(t)) = L(10) 10 L(y(t)) = 3 s L(y(t)) = L(5t2 ) y(t) = 5t2 s[sL(y(t)) − y(0)] − y (0) = L(10) L(y (t)) = L(10) Apply the t-derivative rule to y , that is, replace y by y on page 248. Repeat the t-derivative rule, on y. Use y(0) = y (0) = 0. Use Table 1. Then divide. Apply Table 1, backwards. Invoke Lerch’s cancellation law. Apply L across y = 10.

Existence of the Transform. The Laplace integral is known to exist in the sense of the improper integral
∞ N

∞ −st f (t) dt 0 e definition1

g(t)dt = lim
0

N →∞ 0

g(t)dt

provided f (t) belongs to a class of functions known in the literature as functions of exponential order. For this class of functions the relation (2) f (t) =0 t→∞ eat lim

is required to hold for some real number a, or equivalently, for some constants M and α, (3) |f (t)| ≤ M eαt . In addition, f (t) is required to be piecewise continuous on each finite subinterval of 0 ≤ t < ∞, a term defined as follows.
1 An advanced calculus background is assumed for the Laplace transform existence proof. Applications of Laplace theory require only a calculus background.

250

Laplace Transform

Definition 1 (piecewise continuous) A function f (t) is piecewise continuous on a finite interval [a, b] provided there exists a partition a = t0 < · · · < tn = b of the interval [a, b] and functions f1 , f2 , . . . , fn continuous on (−∞, ∞) such that for t not a partition point . . f (t) = .   fn (t)
  f1 (t) 

t0

(4)

< t < t1 , . . .

tn−1 < t < tn .

The values of f at partition points are undecided by equation (4). In particular, equation (4) implies that f (t) has one-sided limits at each point of a < t < b and appropriate one-sided limits at the endpoints. Therefore, f has at worst a jump discontinuity at each partition point. 3 Example (Exponential order) Show that f (t) = et cos t + t is of exponential order, that is, show that f (t) is piecewise continuous and find α > 0 such that limt→∞ f (t)/eαt = 0. Solution: Already, f (t) is continuous, hence piecewise continuous. From L’Hospital’s rule in calculus, limt→∞ p(t)/eαt = 0 for any polynomial p and any α > 0. Choose α = 2, then t→∞ lim

cos t t f (t) = lim + lim 2t = 0. 2t t→∞ et t→∞ e e

Theorem 1 (Existence of L(f )) Let f (t) be piecewise continuous on every finite interval in t ≥ 0 and satisfy |f (t)| ≤ M eαt for some constants M and α. Then L(f (t)) exists for s > α and lims→∞ L(f (t)) = 0. Proof: It has to be shown that the Laplace integral of f is finite for s > α. Advanced calculus implies that it is sufficient to show that the integrand is absolutely bounded above by an integrable function g(t). Take g(t) = M e −(s−α)t . Then g(t) ≥ 0. Furthermore, g is integrable, because
∞

g(t)dt =
0

M . s−α

Inequality |f (t)| ≤ M eαt implies the absolute value of the Laplace transform integrand f (t)e−st is estimated by f (t)e−st ≤ M eαt e−st = g(t). M , because the s−α right side of this inequality has limit zero at s = ∞. The proof is complete. The limit statement follows from |L(f (t))| ≤
∞ 0

g(t)dt =

7.1 Introduction to the Laplace Method

251

Theorem 2 (Lerch) ∞ If f1 (t) and f2 (t) are continuous, of exponential order and 0 f1 (t)e−st dt = ∞ −st dt for all s > s , then f (t) = f (t) for t ≥ 0. 0 1 2 0 f2 (t)e Proof: See Widder [?]. Theorem 3 (t-Derivative Rule) If f (t) is continuous, lim f (t)e−st = 0 for all large values of s and f (t) t→∞ is piecewise continuous, then L(f (t)) exists for all large s and L(f (t)) = sL(f (t)) − f (0). Proof: See page 276.

Exercises 7.1
Laplace method.
Solve the given 18. f (t) = n=1 cn sin(nt), for any choice of the constants c1 , . . . , cN . initial value problem using Laplace’s method. Existence of transforms. Let f (t) = 1. y = −2, y(0) = 0. 2 2 tet sin(et ). Establish these results. 2. y = 1, y(0) = 0. 19. The function f (t) is not of expo3. y = −t, y(0) = 0. 4. y = t, y(0) = 0. 5. y = 1 − t, y(0) = 0. 6. y = 1 + t, y(0) = 0. 7. y = 3 − 2t, y(0) = 0. 8. y = 3 + 2t, y(0) = 0. 9. y = −2, y(0) = y (0) = 0. 10. y = 1, y(0) = y (0) = 0. 11. y = 1 − t, y(0) = y (0) = 0. 12. y = 1 + t, y(0) = y (0) = 0. 13. y = 3 − 2t, y(0) = y (0) = 0. 14. y = 3 + 2t, y(0) = y (0) = 0. is of exponential order, by finding a constant α ≥ 0 in each case such that f (t) = 0. lim t→∞ eαt 15. f (t) = 1 + t 16. f (t) = et sin(t) nential order. 20. The Laplace integral of f (t), ∞ f (t)e−st dt, converges for all 0 s > 0.
N

Jump Magnitude. For f piecewise continuous, define the jump at t by
J(t) = lim f (t + h) − lim f (t − h). h→0+ h→0+

Compute J(t) for the following f . 21. f (t) = 1 for t ≥ 0, else f (t) = 0 22. f (t) = 1 for t ≥ 1/2, else f (t) = 0 23. f (t) = t/|t| for t = 0, f (0) = 0 24. f (t) = sin t/| sin t| for t = nπ, f (nπ) = (−1)n The series relation ∞ n L( n=0 cn t ) = n=0 cn L(t ) often holds, in which case the result L(tn ) = n!s−1−n can be employed to find a series representation of the Laplace transform. Use this idea on the following to find a series formula for L(f (t)).
∞ n n=0 (2t) /n! ∞ n n=0 (−t) /n!

Exponential order. Show that f (t) Taylor series. ∞ n

2t N 17. f (t) = n=0 cn xn , for any choice 25. f (t) = e = of the constants c0 , . . . , cN . 26. f (t) = e−t =

252

Laplace Transform

7.2 Laplace Integral Table
The objective in developing a table of Laplace integrals, e.g., Tables 4 and 5, is to keep the table size small. Table manipulation rules appearing in Table 6, page 257, effectively increase the table size manyfold, making it possible to solve typical differential equations from electrical and mechanical problems. The combination of Laplace tables plus the table manipulation rules is called the Laplace transform calculus. Table 4 is considered to be a table of minimum size to be memorized. Table 5 adds a number of special-use entries. For instance, the Heaviside entry in Table 5 is memorized, but usually not the others. Derivations are postponed to page 270. The theory of the gamma function Γ(x) appears below on page 255.
Table 4. A minimal Laplace integral table with L-notation
∞ n −st (t )e 0

dt = dt =

n! s1+n 1 s−a s2

L(tn ) = L(eat ) =

n! s1+n 1 s−a s2

∞ at −st 0 (e )e

s + b2 b ∞ −st dt = 2 0 (sin bt)e s + b2
∞ (cos bt)e−st 0

dt =

s + b2 b L(sin bt) = 2 s + b2 L(cos bt) =

Table 5. Laplace integral table extension e−as (a ≥ 0) s

L(H(t − a)) =

L(δ(t − a)) = e−as L(floor(t/a)) = L(sqw(t/a)) = e−as s(1 − e−as ) 1 tanh(as/2) s 1 tanh(as/2) s2

Heaviside unit step, defined by 1 for t ≥ 0, H(t) = 0 otherwise. Dirac delta, δ(t) = dH(t). Special usage rules apply. Staircase function, floor(x) = greatest integer ≤ x. Square wave, sqw(x) = (−1)floor(x) . Triangular wave, x trw(x) = 0 sqw(r)dr. Generalized power function, ∞ Γ(1 + α) = 0 e−x xα dx. Because Γ(1/2) = √ π.

L(a trw(t/a)) = L(tα ) =

Γ(1 + α) s1+α π s

L(t−1/2 ) =

7.2 Laplace Integral Table

253

4 Example (Laplace transform) Let f (t) = t(t − 1) − sin 2t + e3t . Compute L(f (t)) using the basic Laplace table and transform linearity properties. Solution:
L(f (t)) = L(t2 − 5t − sin 2t + e3t )
2

= L(t ) − 5L(t) − L(sin 2t) + L(e ) 2 5 2 1 = 3− 2− 2 + s s s +4 s−3

3t

Expand t(t − 5). Table lookup.

Linearity applied.

5 Example (Inverse Laplace transform) Use the basic Laplace table backwards plus transform linearity properties to solve for f (t) in the equation L(f (t)) = Solution:
L(f (t)) = 1 1 1 2 s +2 + 2+ + 16 s−3 s 2 s3 3t = L(cos 4t) + 2L(e ) + L(t) + 1 L(t2 ) 2 s2
3t 1 2 2t ) 3t 1 2 2t

s2

s 2 s+1 + + 3 . + 16 s − 3 s

Convert to table entries. Laplace table (backwards). Linearity applied. Lerch’s cancellation law.

f (t) = cos 4t + 2e + t +

= L(cos 4t + 2e + t +

6 Example (Heaviside) Find the Laplace transform of f (t) in Figure 1. 5
Figure 1. A piecewise defined function f (t) on 0 ≤ t < ∞: f (t) = 0 except for 1 ≤ t < 2 and 3 ≤ t < 4.

1 1 3 5

Solution: The details require the use of the Heaviside function formula
H(t − a) − H(t − b) = The formula for f (t):   1 1 ≤ t < 2, 5 3 ≤ t < 4, = f (t) =  0 otherwise L(f (t)) = L(f1 (t)) + 5L(f2 (t)) 1 a ≤ t < b, 0 otherwise.

1 0

1 ≤ t < 2, +5 otherwise

1 0

3 ≤ t < 4, otherwise

Then f (t) = f1 (t) + 5f2 (t) where f1 (t) = H(t − 1) − H(t − 2) and f2 (t) = H(t − 3) − H(t − 4). The extended table gives Linearity. Substitute for f1 . = L(H(t − 1)) − L(H(t − 2)) + 5L(f2 (t))

254 e−s − e−2s + 5L(f2 (t)) s e−s − e−2s + 5e−3s − 5e−4s = s

Laplace Transform

=

Extended table used. Similarly for f2 .

7 Example (Dirac delta) A machine shop tool that repeatedly hammers a die is modeled by the Dirac impulse model f (t) = N δ(t − n). Show n=1 that L(f (t)) = N e−ns . n=1 Solution:
L(f (t)) = L = =
N n=1

δ(t − n) − n)) Linearity. Extended Laplace table.

N n=1 L(δ(t N −ns n=1 e

8 Example (Square wave) A periodic camshaft force f (t) applied to a mechanical system has the idealized graph shown in Figure 2. Show that f (t) = 1 + sqw(t) and L(f (t)) = 1 (1 + tanh(s/2)). s 2

0

1

3

Figure 2. A periodic force f (t) applied to a mechanical system.

Solution:
1 + sqw(t) = = 1+1 1−1 2n ≤ t < 2n + 1, n = 0, 1, . . ., 2n + 1 ≤ t < 2n + 2, n = 0, 1, . . ., 2n ≤ t < 2n + 1, n = 0, 1, . . ., otherwise, 1 tanh(s/2) + . s s

2 0 = f (t).

By the extended Laplace table, L(f (t)) = L(1) + L(sqw(t)) =

9 Example (Sawtooth wave) Express the P -periodic sawtooth wave represented in Figure 3 as f (t) = ct/P − c floor(t/P ) and obtain the formula L(f (t)) = c ce−P s c − . P s2 s − se−P s

0

P

4P

Figure 3. A P -periodic sawtooth wave f (t) of height c > 0.

7.2 Laplace Integral Table

255

Solution: The representation originates from geometry, because the periodic function f can be viewed as derived from ct/P by subtracting the correct constant from each of intervals [P, 2P ], [2P, 3P ], etc.
The technique used to verify the identity is to define g(t) = ct/P − c floor(t/P ) and then show that g is P -periodic and f (t) = g(t) on 0 ≤ t < P . Two P periodic functions equal on the base interval 0 ≤ t < P have to be identical, hence the representation follows. The fine details: for 0 ≤ t < P , floor(t/P ) = 0 and floor(t/P + k) = k. Hence g(t + kP ) = ct/P + ck − c floor(k) = ct/P = g(t), which implies that g is P -periodic and g(t) = f (t) for 0 ≤ t < P . L(f (t)) = c L(t) − cL(floor(t/P )) P c ce−P s = − P s2 s − se−P s Linearity. Basic and extended table applied.

10 Example (Triangular wave) Express the triangular wave f of Figure 4 in 5 terms of the square wave sqw and obtain L(f (t)) = 2 tanh(πs/2). πs 5

0

2π

Figure 4. A 2π-periodic triangular wave f (t) of height 5. t/π 0

Solution: The representation of f in terms of sqw is f (t) = 5

sqw(x)dx.

Details: A 2-periodic triangular wave of height 1 is obtained by integrating the square wave of period 2. A wave of height c and period 2 is given by t 2t/P c trw(t) = c 0 sqw(x)dx. Then f (t) = c trw(2t/P ) = c 0 sqw(x)dx where c = 5 and P = 2π. Laplace transform details: Use the extended Laplace table as follows. L(f (t)) = 5 5 L(π trw(t/π)) = 2 tanh(πs/2). π πs

Gamma Function. In mathematical physics, the Gamma function or the generalized factorial function is given by the identity
∞

(1)

Γ(x) =
0

e−t tx−1 dt,

x > 0.

This function is tabulated and available in computer languages like Fortran, C and C++. It is also available in computer algebra systems and numerical laboratories. Some useful properties of Γ(x): (2) (3) Γ(1 + x) = xΓ(x) Γ(1 + n) = n! for integers n ≥ 1.

256 Details for relations (2) and (3): Start with

Laplace Transform
∞ −t e dt 0

= 1, which gives Γ(1) = 1. Use this identity and successively relation (2) to obtain relation (3). To prove identity (2), integration by parts is applied, as follows: Γ(1 + x) = = =
∞ −t x 0 e t dt t=∞ ∞ −tx e−t |t=0 + 0 e−t xtx−1 dt ∞ x 0 e−t tx−1 dt

Definition. Use u = tx , dv = e−t dt. Boundary terms are zero for x > 0.

= xΓ(x).

Exercises 7.2 transform. Compute Inverse Laplace transform. Solve L(f (t)) using the basic Laplace table the given equation for the function and the linearity properties of the f (t). Use the basic table and linearity transform. Do not use the direct properties of the Laplace transform. Laplace transform! 21. L(f (t)) = s−2 1. L(2t) 22. L(f (t)) = 4s−2 2. L(4t) 23. L(f (t)) = 1/s + 2/s2 + 3/s3 2 3. L(1 + 2t + t ) 24. L(f (t)) = 1/s3 + 1/s 2 4. L(t − 3t + 10) 25. L(f (t)) = 2/(s2 + 4) 5. L(sin 2t) 26. L(f (t)) = s/(s2 + 4) 6. L(cos 2t) 27. L(f (t)) = 1/(s − 3) 7. L(e2t ) 28. L(f (t)) = 1/(s + 3) 8. L(e−2t ) 29. L(f (t)) = 1/s + s/(s2 + 4) 9. L(t + sin 2t) 30. L(f (t)) = 2/s − 2/(s2 + 4) 10. L(t − cos 2t) 31. L(f (t)) = 1/s + 1/(s − 3) 11. L(t + e2t ) 32. L(f (t)) = 1/s − 3/(s − 2) 12. L(t − 3e−2t ) 33. L(f (t)) = (2 + s)2 /s3 13. L((t + 1)2 ) 34. L(f (t)) = (s + 1)/s2 14. L((t + 2)2 ) 35. L(f (t)) = s(1/s2 + 2/s3 ) 15. L(t(t + 1)) 36. L(f (t)) = (s + 1)(s − 1)/s3 16. L((t + 1)(t + 2)) 10 37. L(f (t)) = n=0 n!/s1+n 10 n 17. L( n=0 t /n!) 10 38. L(f (t)) = n=0 n!/s2+n 10 n+1 18. L( n=0 t /n!) n 10 39. L(f (t)) = n=1 2 10 s + n2 19. L( n=1 sin nt) s 10 10 40. L(f (t)) = n=0 2 20. L( n=0 cos nt) s + n2 Laplace

7.3 Laplace Transform Rules

257

7.3 Laplace Transform Rules
In Table 6, the basic table manipulation rules are summarized. Full statements and proofs of the rules appear in section 7.7, page 275. The rules are applied here to several key examples. Partial fraction expansions do not appear here, but in section 7.4, in connection with Heaviside’s coverup method.
Table 6. Laplace transform rules L(f (t) + g(t)) = L(f (t)) + L(g(t)) L(cf (t)) = cL(f (t)) L(y (t)) = sL(y(t)) − y(0) 1 L g(x)dx = L(g(t)) s d L(tf (t)) = − L(f (t)) ds t 0

Linearity.
The Laplace of a sum is the sum of the Laplaces.

Linearity.
Constants move through the L-symbol.

The t-derivative rule.
Derivatives L(y ) are replaced in transformed equations.

The t-integral rule. The s-differentiation rule.
Multiplying f by t applies −d/ds to the transform of f .

L(eat f (t)) = L(f (t))|s→(s−a) L(f (t − a)H(t − a)) = e−as L(f (t)), L(g(t)H(t − a)) = e−as L(g(t + a)) L(f (t)) =
P 0

First shifting rule.
Multiplying f by eat replaces s by s − a.

Second shifting rule.
First and second forms.

f (t)e−st dt 1 − e−P s

Rule for P -periodic functions.
Assumed here is f (t + P ) = f (t).

L(f (t))L(g(t)) = L((f ∗ g)(t))

Convolution rule. t Define (f ∗ g)(t) =

0

f (x)g(t − x)dx.

11 Example (Harmonic oscillator) Solve by Laplace’s method the initial value problem x + x = 0, x(0) = 0, x (0) = 1. Solution: The solution is x(t) = sin t. The details: sL(x ) − x (0) + L(x) = 0
2

L(x ) + L(x) = L(0)

Apply L across the equation. Use the t-derivative rule. Use again the t-derivative rule. Use x(0) = 0, x (0) = 1. Divide. Basic Laplace table. Invoke Lerch’s cancellation law.

s[sL(x) − x(0)] − x (0) + L(x) = 0 (s + 1)L(x) = 1 1 L(x) = 2 s +1 = L(sin t) x(t) = sin t

258

Laplace Transform 2 . (s − 5)3

12 Example (s-differentiation rule) Show the steps for L(t2 e5t ) = Solution:
L(t2 e5t ) = − d ds − d ds L(e5t )

Apply s-differentiation. Basic Laplace table. Calculus power rule. Identity verified.

= (−1)2 = = d ds

2 (s − 5)3

1 d d ds ds s − 5 −1 (s − 5)2

13 Example (First shifting rule) Show the steps for L(t2 e−3t ) = Solution:
L(t2 e−3t ) = L(t2 ) = = s→s−(−3) 2 . (s + 3)3

First shifting rule. Basic Laplace table.

2 s2+1 2 (s + 3)3

s→s−(−3)

Identity verified.

14 Example (Second shifting rule) Show the steps for L(sin t H(t − π)) = e−πs . s2 + 1

Solution: The second shifting rule is applied as follows.
L(sin t H(t − π)) = L(g(t)H(t − a) = e−as L(g(t + a) =e
−πs −πs

Choose g(t) = sin t, a = π. Second form, second shifting theorem. Sum rule sin(a + b) = sin a cos b + sin b cos a plus sin π = 0, cos π = −1. Basic Laplace table. Identity verified.

=e

L(− sin t) −1 s2 + 1

L(sin(t + π)) Substitute a = π.

= e−πs

15 Example (Trigonometric formulas) Show the steps used to obtain these Laplace identities: s 2 − a2 2(s3 − 3sa2 ) (a) L(t cos at) = 2 (c) L(t2 cos at) = (s + a2 )2 (s2 + a2 )3 6s2 a − a3 2sa (d) L(t2 sin at) = 2 (b) L(t sin at) = 2 (s + a2 )2 (s + a2 )3

7.3 Laplace Transform Rules Solution: The details for (a):
L(t cos at) = −(d/ds)L(cos at) s d =− 2 + a2 ds s = s2 − a 2 (s2 + a2 )2 Use s-differentiation. Basic Laplace table.

259

Calculus quotient rule.

The details for (c): L(t2 cos at) = −(d/ds)L((−t) cos at) d = ds = s2 − a 2 − 2 (s + a2 )2 Use s-differentiation. Result of (a). Calculus quotient rule.

2s3 − 6sa2 ) (s2 + a2 )3

The similar details for (b) and (d) are left as exercises.

16 Example (Exponentials) Show the steps used to obtain these Laplace identities: (s − a)2 − b2 s−a (c) L(teat cos bt) = (a) L(eat cos bt) = (s − a)2 + b2 ((s − a)2 + b2 )2 2b(s − a) b (d) L(teat sin bt) = (b) L(eat sin bt) = (s − a)2 + b2 ((s − a)2 + b2 )2 Solution: Details for (a):
L(eat cos bt) = L(cos bt)|s→s−a = First shifting rule. Basic Laplace table. Verified (a). s s2 + b2 s→s−a s−a = (s − a)2 + b2

Details for (c): L(teat cos bt) = L(t cos bt)|s→s−a = = = = − − d L(cos bt) ds d ds s→s−a First shifting rule. Apply s-differentiation. Basic Laplace table. s→s−a s s2 + b 2

s2 − b 2 (s2 + b2 )2

Calculus quotient rule. s→s−a (s − a)2 − b2 ((s − a)2 + b2 )2

Verified (c).

Left as exercises are (b) and (d).

260

Laplace Transform

17 Example (Hyperbolic functions) Establish these Laplace transform facts about cosh u = (eu + e−u )/2 and sinh u = (eu − e−u )/2. (a) L(cosh at) = s 2 − a2 s a (b) L(sinh at) = 2 s − a2 Solution: The details for (a):
L(cosh at) = 1 (L(eat ) + L(e−at )) 2 1 = 2 1 1 + s−a s+a s = 2 s − a2 d a 2 − a2 ds s a(2s) = 2 (s − a2 )2 Definition plus linearity of L. Basic Laplace table. Identity (a) verified.

(c) L(t cosh at) =

s 2 + a2 (s2 − a2 )2 2as (d) L(t sinh at) = 2 (s − a2 )2

The details for (d): L(t sinh at) = − Apply the s-differentiation rule. Calculus power rule; (d) verified.

Left as exercises are (b) and (c).

18 Example (s-differentiation) Solve L(f (t)) =

(s2

2s for f (t). + 1)2

Solution: The solution is f (t) = t sin t. The details:
L(f (t)) = 2s (s2 + 1)2 d 1 =− ds s2 + 1 d = − (L(sin t)) ds = L(t sin t)

Calculus power rule (un ) = nun−1 u . Basic Laplace table. Apply the s-differentiation rule. Lerch’s cancellation law.

f (t) = t sin t

19 Example (First shift rule) Solve L(f (t)) =

s+2 for f (t). 22 + 2s + 2

Solution: The answer is f (t) = e−t cos t + e−t sin t. The details:
L(f (t)) = = s+2 s2 + 2s + 2 s+2 (s + 1)2 + 1 Signal for this method: the denominator has complex roots. Complete the square, denominator.

7.3 Laplace Transform Rules
S+1 S2 + 1 1 S + = 2 S + 1 S2 + 1 = L(cos t) + L(sin t)|s→S=s+1 = f (t) = e = L(e
−t −t

261
Substitute S for s + 1. Split into Laplace table entries. Basic Laplace table. First shift rule. Invoke Lerch’s cancellation law.

cos t + e

cos t) + L(e
−t

−t

sin t)

sin t

20 Example (Damped oscillator) Solve by Laplace’s method the initial value problem x + 2x + 2x = 0, x(0) = 1, x (0) = −1. Solution: The solution is x(t) = e−t cos t. The details:
L(x ) + 2L(x ) + 2L(x) = L(0) s[sL(x) − x(0)] − x (0) +2[L(x) − x(0)] + 2L(x) = 0 sL(x ) − x (0) + 2L(x ) + 2L(x) = 0 Apply L across the equation. The t-derivative rule on x . The t-derivative rule on x. Use x(0) = 1, x (0) = −1. Divide. Complete the square in the denominator. Basic Laplace table. First shifting rule. Invoke Lerch’s cancellation law.

(s2 + 2s + 2)L(x) = 1 + s s+1 L(x) = 2 s + 2s + 2 s+1 = (s + 1)2 + 1 = L(cos t)|s→s+1 x(t) = e−t cos t = L(e−t cos t)

21 Example (Rectified sine wave) Compute the Laplace transform of the rectified sine wave f (t) = | sin ωt| and show it can be expressed in the form πs ω coth 2ω . L(| sin ωt|) = s2 + ω 2 Solution: The periodic function formula will be applied with period P =
2π/ω. The calculation reduces to the evaluation of J = 0 f (t)e−st dt. Because sin ωt ≤ 0 on π/ω ≤ t ≤ 2π/ω, integral J can be written as J = J1 + J2 , where π/ω 2π/ω P

J1 =
0

sin ωt e−st dt,

J2 = π/ω − sin ωt e−st dt.

Integral tables give the result sin ωt e−st dt = − Then J1 = ωe−st cos(ωt) se−st sin(ωt) − . s2 + ω 2 s2 + ω 2 ω(e−2πs/ω + e−πs/ω ) , s2 + ω 2

ω(e−π∗s/ω + 1) , s2 + ω 2

J2 =

262 ω(e−πs/ω + 1)2 . s2 + ω 2

Laplace Transform

J=

The remaining challenge is to write the answer for L(f (t)) in terms of coth. The details: L(f (t)) = = = = J 1 − e−P s (1 − Periodic function formula. J + e−P s/2 ) Apply 1 − x2 = (1 − x)(1 + x), x = e−P s/2 . Cancel factor 1 + e−P s/2 . Factor out e−P s/4 , then cancel. Apply cosh, sinh identities. Use coth u = cosh u/ sinh u. Identity verified.

e−P s/2 )(1

eP s/4 + e−P s/4 ω eP s/4 − e−P s/4 s2 + ω 2 2 cosh(P s/4) ω = 2 + ω2 2 sinh(P s/4) s ω coth(P s/4) = s2 + ω 2 πs ω coth 2ω = s2 + ω 2

ω(1 + e−P s/2 ) (1 − e−P s/2 )(s2 + ω 2 )

22 Example (Half–wave rectification) Compute the Laplace transform of the half–wave rectification of sin ωt, denoted g(t), in which the negative cycles of sin ωt have been canceled to create g(t). Show in particular that L(g(t)) = 1 ω πs 1 + coth 2 + ω2 2s 2ω

Solution: The half–wave rectification of sin ωt is g(t) = (sin ωt + | sin ωt|)/2. Therefore, the basic Laplace table plus the result of Example 21 give
L(2g(t)) = L(sin ωt) + L(| sin ωt|) ω ω cosh(πs/(2ω)) = 2 + 2 s +ω s2 + ω 2 ω (1 + cosh(πs/(2ω)) = 2 s + ω2 Dividing by 2 produces the identity.

23 Example (Shifting rules) Solve L(f (t)) = e−3s

s2

s+1 for f (t). + 2s + 2

Solution: The answer is f (t) = e3−t cos(t − 3)H(t − 3). The details:
L(f (t)) = e−3s s+1 (s + 1)2 + 1 S = e−3s 2 S +1 = e−3S+3 (L(cos t))|s→S=s+1 Complete the square. Replace s + 1 by S. Basic Laplace table.

7.3 Laplace Transform Rules
= e3 e−3s L(cos t) = e L(e
3−t 3 −t

263
Regroup factor e−3S . Second shifting rule. First shifting rule. Lerch’s cancellation law.

s→S=s+1

= e3 (L(cos(t − 3)H(t − 3)))|s→S=s+1 cos(t − 3)H(t − 3) cos(t − 3)H(t − 3))

f (t) = e

24 Example () Solve L(f (t) =

s2

s+7 for f (t). + 4s + 8

Solution: The answer is f (t) = e−2t (cos 2t + 5 sin 2t). The details: 2
L(f (t)) = s+7 (s + 2)2 + 4 S+5 = 2 S +4 5 2 S + = 2 S + 4 2 S2 + 4 s 5 2 = 2 + s + 4 2 s2 + 4 s→S=s+2 s→S=s+2 −2t

Complete the square. Replace s + 2 by S. Split into table entries. Prepare for shifting rule. Basic Laplace table. First shifting rule. Lerch’s cancellation law.

5 = L(cos 2t) + 2 L(sin 2t)

f (t) = e

= L(e
−2t

(cos 2t +
5 2

5 2

sin 2t))

(cos 2t +

sin 2t)

264

Laplace Transform

7.4 Heaviside’s Method
This practical method was popularized by the English electrical engineer Oliver Heaviside (1850–1925). A typical application of the method is to solve 2s = L(f (t)) (s + 1)(s2 + 1) for the t-expression f (t) = −e−t + cos t + sin t. The details in Heaviside’s method involve a sequence of easy-to-learn college algebra steps. More precisely, Heaviside’s method systematically converts a polynomial quotient a0 + a1 s + · · · + an sn (1) b0 + b1 s + · · · + bm sm into the form L(f (t)) for some expression f (t). It is assumed that a0 , .., an , b0 , . . . , bm are constants and the polynomial quotient (1) has limit zero at s = ∞.

Partial Fraction Theory
In college algebra, it is shown that a rational function (1) can be expressed as the sum of terms of the form (2) A (s − s0 )k

where A is a real or complex constant and (s − s0 )k divides the denominator in (1). In particular, s0 is a root of the denominator in (1). Assume fraction (1) has real coefficients. If s0 in (2) is real, then A is real. If s0 = α + iβ in (2) is complex, then (s − s0 )k also appears, where s0 = α − iβ is the complex conjugate of s0 . The corresponding terms in (2) turn out to be complex conjugates of one another, which can be combined in terms of real numbers B and C as (3) B+Cs A A + = . k k (s − s0 ) (s − s0 ) ((s − α)2 + β 2 )k

Simple Roots. Assume that (1) has real coefficients and the denominator of the fraction (1) has distinct real roots s1 , . . . , sN and distinct complex roots α1 + iβ1 , . . . , αM + iβM . The partial fraction expansion of (1) is a sum given in terms of real constants Ap , Bq , Cq by
(4)
N M Ap Bq + Cq (s − αq ) a0 + a1 s + · · · + an sn = + . m 2 b0 + b1 s + · · · + bm s s − sp q=1 (s − αq )2 + βq p=1

7.4 Heaviside’s Method

265

Multiple Roots. Assume (1) has real coefficients and the denominator of the fraction (1) has possibly multiple roots. Let Np be the multiplicity of real root sp and let Mq be the multiplicity of complex root αq + iβq , 1 ≤ p ≤ N , 1 ≤ q ≤ M . The partial fraction expansion of (1) is given in terms of real constants Ap,k , Bq,k , Cq,k by
N

(5) p=1 1≤k≤Np

M Ap,k Bq,k + Cq,k (s − αq ) + . 2 (s − sp )k q=1 1≤k≤M ((s − αq )2 + βq )k q Heaviside’s Coverup Method
The method applies only to the case of distinct roots of the denominator in (1). Extensions to multiple-root cases can be made; see page 266. To illustrate Oliver Heaviside’s ideas, consider the problem details (6) 2s + 1 s(s − 1)(s + 1) = A B C + + s s−1 s+1

= L(A) + L(Bet ) + L(Ce−t ) = L(A + Bet + Ce−t ) The first line (6) uses college algebra partial fractions. The second and third lines use the Laplace integral table and properties of L.

Heaviside’s mysterious method. Oliver Heaviside proposed to find in (6) the constant C =
1 2

by a cover–up method: = s+1 =0

2s + 1 s(s − 1)

C

.

The instructions are to cover–up the matching factors (s + 1) on the left , then evaluate on the left at the root s which and right with box makes the contents of the box zero. The other terms on the right are replaced by zero. To justify Heaviside’s cover–up method, multiply (6) by the denominator s + 1 of partial fraction C/(s + 1): (2s + 1) (s + 1) s(s − 1) (s + 1) = A (s + 1) s + B (s + 1) s−1 + C (s + 1) (s + 1) .

Set (s + 1) = 0 in the display. Cancellations left and right plus annihilation of two terms on the right gives Heaviside’s prescription 2s + 1 s(s − 1) = C. s+1=0 266

Laplace Transform

The factor (s + 1) in (6) is by no means special: the same procedure applies to find A and B. The method works for denominators with simple roots, that is, no repeated roots are allowed.

Extension to Multiple Roots. An extension of Heaviside’s method is possible for the case of repeated roots. The basic idea is to factor–out the repeats. To illustrate, consider the partial fraction expansion details R= = 1 (s + 1)2 (s + 2) 1 (s + 1)(s + 2) −1 1 + s+1 s+2 A sample rational function having repeated roots. Factor–out the repeats. Apply the cover–up method to the simple root fraction. Multiply. Apply the cover–up method to the last fraction on the right.

1 s+1 1 = s+1 =

−1 1 + 2 (s + 1) (s + 1)(s + 2) 1 −1 1 = + + (s + 1)2 s+1 s+2

Terms with only one root in the denominator are already partial fractions. Thus the work centers on expansion of quotients in which the denominator has two or more roots.

Special Methods. Heaviside’s method has a useful extension for the case of roots of multiplicity two. To illustrate, consider these details: R= 1 (s + 1)2 (s + 2) C A B + = + 2 s + 1 (s + 1) s+2 1 A 1 + = + 2 s + 1 (s + 1) s+2 = 1 −1 1 + + s + 1 (s + 1)2 s + 2 A fraction with multiple roots. See equation (5). Find B and C by Heaviside’s cover– up method. Multiply by s+1. Set s = ∞. Then 0 = A + 1.

The illustration works for one root of multiplicity two, because s = ∞ will resolve the coefficient not found by the cover–up method. In general, if the denominator in (1) has a root s0 of multiplicity k, then the partial fraction expansion contains terms A2 Ak A1 + + ··· + . s − s0 (s − s0 )2 (s − s0 )k Heaviside’s cover–up method directly finds Ak , but not A1 to Ak−1 .

7.5 Heaviside Step and Dirac Delta

267

7.5 Heaviside Step and Dirac Delta
Heaviside Function. The unit step function or Heaviside function is defined by H(x) = 1 0 for x ≥ 0, for x < 0.

The most often–used formula involving the Heaviside function is the characteristic function of the interval a ≤ t < b, given by (1) H(t − a) − H(t − b) = 1 a ≤ t < b, 0 t < a, t ≥ b.

To illustrate, a square wave sqw(t) = (−1)floor(t) can be written in the series form
∞

(−1)n (H(t − n) − H(t − n − 1)). n=0 Dirac Delta. A precise mathematical definition of the Dirac delta, denoted δ, is not possible to give here. Following its inventor P. Dirac, the definition should be δ(t) = dH(t). The latter is nonsensical, because the unit step does not have a calculus derivative at t = 0. However, dH(t) could have the meaning of a Riemann-Stieltjes integrator, which restrains dH(t) to have meaning only under an integral sign. It is in this sense that the Dirac delta δ is defined. What do we mean by the differential equation x + 16x = 5δ(t − t0 )? The equation x + 16x = f (t) represents a spring-mass system without damping having Hooke’s constant 16, subject to external force f (t). In a mechanical context, the Dirac delta term 5δ(t − t0 ) is an idealization of a hammer-hit at time t = t0 > 0 with impulse 5. More precisely, the forcing term f (t) can be formally written as a RiemannStieltjes integrator 5dH(t−t0 ) where H is Heaviside’s unit step function. The Dirac delta or “derivative of the Heaviside unit step,” nonsensical as it may appear, is realized in applications via the two-sided or central difference quotient H(t + h) − H(t − h) ≈ dH(t). 2h

268

Laplace Transform

Therefore, the force f (t) in the idealization 5δ(t − t0 ) is given for h > 0 very small by the approximation f (t) ≈ 5 H(t − t0 + h) − H(t − t0 − h) . 2h

The impulse2 of the approximated force over a large interval [a, b] is computed from b h

f (t)dt ≈ 5 a −h

H(t − t0 + h) − H(t − t0 − h) dt = 5, 2h

due to the integrand being 1/(2h) on |t − t0 | < h and otherwise 0.

Modeling Impulses. One argument for the Dirac delta idealization is that an infinity of choices exist for modeling an impulse. There are in addition to the central difference quotient two other popular difference quotients, the forward quotient (H(t + h) − H(t))/h and the backward quotient (H(t) − H(t − h))/h (h > 0 assumed). In reality, h is unknown in any application, and the impulsive force of a hammer hit is hardly constant, as is supposed by this naive modeling. The modeling logic often applied for the Dirac delta is that the external force f (t) is used in the model in a limited manner, in which only the momentum p = mv is important. More precisely, only the change in b momentum or impulse is important, a f (t)dt = ∆p = mv(b) − mv(a). The precise force f (t) is replaced during the modeling by a simplistic piecewise-defined force that has exactly the same impulse ∆p. The replacement is justified by arguing that if only the impulse is important, and not the actual details of the force, then both models should give similar results. Function or Operator? The work of physics Nobel prize winner P. Dirac (1902–1984) proceeded for about 20 years before the mathematical community developed a sound mathematical theory for his impulsive force representations. A systematic theory was developed in 1936 by the soviet mathematician S. Sobolev. The French mathematician L. Schwartz further developed the theory in 1945. He observed that the idealization is not a function but an operator or linear functional, in particular, δ maps or associates to each function φ(t) its value at t = 0, in short, δ(φ) = φ(0). This fact was observed early on by Dirac and others, during the replacement of simplistic forces by δ. In Laplace theory, there is a natural encounter with the ideas, because L(f (t)) routinely appears on the right of the equation after transformation. This term, in the case
Momentum is defined to be mass times velocity. If the force f is given by Newton’s b d law as f (t) = dt (mv(t)) and v(t) is velocity, then a f (t)dt = mv(b) − mv(a) is the net momentum or impulse.
2

7.5 Heaviside Step and Dirac Delta

269

of an impulsive force f (t) = c(H(t−t0 −h)−H(t−t0 +h))/(2h), evaluates for t0 > 0 and t0 − h > 0 as follows: L(f (t)) = c (H(t − t0 − h) − H(t − t0 + h))e−st dt 2h 0 t0 +h c = e−st dt t0 −h 2h esh − e−sh = ce−st0 2sh
∞

esh − e−sh The factor is approximately 1 for h > 0 small, because of 2sh L’Hospital’s rule. The immediate conclusion is that we should replace the impulsive force f by an equivalent one f ∗ such that L(f ∗ (t)) = ce−st0 . Well, there is no such function f ∗ ! The apparent mathematical flaw in this idea was resolved by the work of L. Schwartz on distributions. In short, there is a solid foundation for introducing f ∗ , but unfortunately the mathematics involved is not elementary nor especially accessible to those readers whose background is just calculus. Practising engineers and scientists might be able to ignore the vast literature on distributions, citing the example of physicist P. Dirac, who succeeded in applying impulsive force ideas without the distribution theory developed by S. Sobolev and L. Schwartz. This will not be the case for those who wish to read current literature on partial differential equations, because the work on distributions has forever changed the required background for that topic.

270

Laplace Transform

7.6 Laplace Table Derivations
Verified here are two Laplace tables, the minimal Laplace Table 7.2-4 and its extension Table 7.2-5. Largely, this section is for reading, as it is designed to enrich lectures and to aid readers who study alone. Derivation of Laplace integral formulas in Table 7.2-4, page 252.
The first step is to evaluate L(tn ) for n = 0. L(1) = =
∞ (1)e−st dt 0 t=∞ −(1/s)e−st |t=0

• Proof of L(tn ) = n!/s1+n:

Laplace integral of f (t) = 1. Evaluate the integral. Assumed s > 0 to evaluate limt→∞ e−st .

= 1/s

The value of L(tn ) for n = 1 can be obtained by s-differentiation of the relation L(1) = 1/s, as follows. d ds L(1)

= = =

∞ d −st dt ds 0 (1)e ∞ d −st ) dt 0 ds (e ∞ −st (−t)e dt 0

Laplace integral for f (t) = 1. Used d ds b a

F dt = u b dF a ds

dt.

Calculus rule (e ) = u eu . Definition of L(t).

= −L(t) Then d L(t) = − ds L(1)

Rewrite last display. Use L(1) = 1/s. Differentiate.

=

= 1/s

d − ds (1/s) 2

d This idea can be repeated to give L(t2 ) = − ds L(t) and hence L(t2 ) = 2/s3 . d n n−1 The pattern is L(t ) = − ds L(t ) which gives L(tn ) = n!/s1+n .

The result follows from L(1) = 1/s, as follows. L(eat ) = = =
∞ at −st dt 0 e e ∞ −(s−a)t e dt 0 ∞ −St dt 0 e

• Proof of L(eat ) = 1/(s − a):

Direct Laplace transform. Use eA eB = eA+B . Apply L(1) = 1/s. Substitute S = s − a.

= 1/S = 1/(s − a)

Back-substitute S = s − a.

• Proof of L(cos bt) = s/(ss + b2 ) and L(sin bt) = b/(ss + b2 ):

Use will be made of Euler’s formula eiθ = cos θ + i sin θ, usually first introduced √ in trigonometry. In this formula, θ is a real number (in radians) and i = −1 is the complex unit.

7.6 Laplace Table Derivations eibt e−st = (cos bt)e−st + i(sin bt)e−st
∞ −ibt −st e e dt 0

271
Substitute θ = bt into Euler’s formula and multiply by e−st . Integrate t = 0 to t = ∞. Use properties of integrals. Evaluate the left side using L(eat ) = 1/(s − a), a = ib. Direct Laplace transform definition. Use complex rule 1/z = z/|z|2 , z = A + iB, z = A − iB, |z| = √ A2 + B 2 . Extract the real part. Extract the imaginary part.

=

∞ (cos bt)e−st dt 0 ∞ + i 0 (sin bt)e−st dt

1 = s − ib

∞ (cos bt)e−st dt 0 ∞ + i 0 (sin bt)e−st dt

1 = L(cos bt) + iL(sin bt) s − ib

s + ib = L(cos bt) + iL(sin bt) s2 + b 2 s = L(cos bt) s2 + b 2 b = L(sin bt) 2 + b2 s

Derivation of Laplace integral formulas in Table 7.2-5, page 252.

• Proof of the Heaviside formula L(H(t − a)) = e−as /s.
L(H(t − a)) = = = =
∞ H(t − a)e−st dt 0 ∞ (1)e−st dt a ∞ −s(x+a) dx 0 (1)e −as ∞ −sx e (1)e dx 0 −as

Because H(t − a) = 0 for 0 ≤ t < a. Change variables t = x + a.

Direct Laplace transform. Assume a ≥ 0.

Constant e−as moves outside integral. Apply L(1) = 1/s.

=e

(1/s)

The definition of the delta function is a formal one, in which every occurrence of δ(t − a)dt under an integrand is replaced by dH(t − a). The differential symbol dH(t − a) is taken in the sense of the Riemann-Stieltjes integral. This integral is defined in [?] for monotonic integrators α(x) as the limit b N

• Proof of the Dirac delta formula L(δ(t − a)) = e−as .

f (x)dα(x) = lim a N →∞

n=1

f (xn )(α(xn ) − α(xn−1 ))

The steps in computing the Laplace integral of the delta function appear below. Admittedly, the proof requires advanced calculus skills and a certain level of mathematical maturity. The reward is a fuller understanding of the Dirac symbol δ(x). L(δ(t − a)) =
∞ −st e δ(t − a)dt 0 ∞ −st e dH(t − a) 0 M limM→∞ 0 e−st dH(t

where x0 = a, xN = b and x0 < x1 < · · · < xN forms a partition of [a, b] whose mesh approaches zero as N → ∞.

Laplace integral, a > 0 assumed. Replace δ(t − a)dt by dH(t − a). − a) Definition of improper integral.

=

=

272
= e−sa

Laplace Transform
Explained below.

To explain the last step, apply the definition of the Riemann-Stieltjes integral:
M 0 N −1

e−st dH(t − a) = lim

N →∞

n=0

e−stn (H(tn − a) − H(tn−1 − a))

where 0 = t0 < t1 < · · · < tN = M is a partition of [0, M ] whose mesh max1≤n≤N (tn − tn−1 ) approaches zero as N → ∞. Given a partition, if tn−1 < a ≤ tn , then H(tn −a)−H(tn−1 −a) = 1, otherwise this factor is zero. Therefore, the sum reduces to a single term e−stn . This term approaches e−sa as N → ∞, because tn must approach a. −as • Proof of L(floor(t/a)) = s(1 e e−as ) : − The library function floor present in computer languages C and Fortran is defined by floor(x) = greatest whole integer ≤ x, e.g., floor(5.2) = 5 and floor(−1.9) = −2. The computation of the Laplace integral of floor(t) requires ideas from infinite series, as follows. F (s) = = = = = = = = =
∞ floor(t)e−st dt 0 n+1 ∞ (n)e−st dt n=0 n ∞ n −ns (e − e−ns−s ) n=0

Laplace integral definition. On n ≤ t < n + 1, floor(t) = n. Evaluate each integral. Common factor removed. Define x = e−s . Term-by-term differentiation. Geometric series sum. Compute the derivative, simplify. Substitute x = e−s .

s

1 − e−s ∞ −sn n=0 ne s x(1 − x) ∞ n−1 n=0 nx s x(1 − x) d ∞ xn s dx n=0 x(1 − x) d 1 s dx 1 − x x s(1 − x) e−s s(1 − e−s )
∞ −st dt 0 floor(t/a)e ∞ −asr a 0 floor(r)e dr

To evaluate the Laplace integral of floor(t/a), a change of variables is made. L(floor(t/a)) = Laplace integral definition. Change variables t = ar. Apply the formula for F (s). Simplify.

=

= aF (as) e−as = s(1 − e−as )

• Proof of L(sqw(t/a)) = 1 tanh(as/2): s
The square wave defined by sqw(x) = (−1)floor(x) is periodic of period 2 and 2 piecewise-defined. Let P = 0 sqw(t)e−st dt.

7.6 Laplace Table Derivations
P= = =
1 2 sqw(t)e−st dt + 1 sqw(t)e−st dt 0 1 −st 2 e dt − 1 e−st dt 0

273
Apply
b a

=

c b + c. a

1 1 (1 − e−s ) + (e−2s − e−s ) s s 1 = (1 − e−s )2 s

Use sqw(x) = 1 on 0 ≤ x < 1 and sqw(x) = −1 on 1 ≤ x < 2. Evaluate each integral. Collect terms.

An intermediate step is to compute the Laplace integral of sqw(t): L(sqw(t)) = = = = = = sqw(t)e−st dt 1 − e−2s 1 1 (1 − e−s )2 . s 1 − e−2s 1 1 − e−s . s 1 + e−s 1 es/2 − e−s/2 . s es/2 + e−s/2 1 sinh(s/2) . s cosh(s/2) 1 tanh(s/2). s
2 0

Periodic function formula, page 275. Use the computation of P above. Factor 1 − e−2s = (1 − e−s )(1 + e−s ). Multiply the fraction by es/2 /es/2 . Use sinh u = (eu − e−u )/2, cosh u = (eu + e−u )/2. Use tanh u = sinh u/ cosh u.

To complete the computation of L(sqw(t/a)), a change of variables is made: L(sqw(t/a)) = sqw(r)e (a)dr a = tanh(as/2) as 1 = tanh(as/2) s =
∞ 0 ∞ 0

sqw(t/a)e−st dt
−asr

Direct transform. Change variables r = t/a. See L(sqw(t)) above.

• Proof of L(a trw(t/a)) = s12 tanh(as/2):
The triangular wave is defined by trw(t) = L(a trw(t/a)) = = = 1 (f (0) + L(f (t)) s 1 L(sqw(t/a)) s 1 tanh(as/2) s2 t 0

sqw(x)dx.

Let f (t) = a trw(t/a). Use L(f (t)) = sL(f (t)) − f (0), page 251. Use f (0) = 0, (a sqw(t/a). Table entry for sqw. t/a 0

sqw(x)dx) =

+ • Proof of L(tα ) = Γ(11+αα) : s L(tα ) =
∞ α −st t e dt 0 ∞ α −u du/s 0 (u/s) e

Direct Laplace transform. Change variables u = st, du = sdt.

=

274
1 ∞ α −u u e du s1+α 0 1 = 1+α Γ(1 + α). s =

Laplace Transform

Where Γ(x) = definition.

∞ 0

ux−1 e−u du, by

The generalized factorial function Γ(x) is defined for x > 0 and it agrees with the classical factorial n! = (1)(2) · · · (n) in case x = n + 1 is an integer. In literature, α! means Γ(1 + α). For more details about the Gamma function, see Abramowitz and Stegun [?], or maple documentation.

• Proof of L(t−1/2 ) =
L(t−1/2 ) =

π : s Apply the previous formula. Use Γ(1/2) = √ π.

Γ(1 + (−1/2)) s1−1/2 √ π = √ s

7.7 Transform Properties

275

7.7 Transform Properties
Collected here are the major theorems and their proofs for the manipulation of Laplace transform tables. Theorem 4 (Linearity) The Laplace transform has these inherited integral properties: (a) (b) L(f (t) + g(t)) = L(f (t)) + L(g(t)), L(cf (t)) = cL(f (t)).

Theorem 5 (The t-Derivative Rule) Let y(t) be continuous, of exponential order and let f (t) be piecewise continuous on t ≥ 0. Then L(y (t)) exists and L(y (t)) = sL(y(t)) − y(0). Theorem 6 (The t-Integral Rule) Let g(t) be of exponential order and continuous for t ≥ 0. Then L t 0 g(x) dx

1 = L(g(t)). s

Theorem 7 (The s-Differentiation Rule) Let f (t) be of exponential order. Then L(tf (t)) = − d L(f (t)). ds

Theorem 8 (First Shifting Rule) Let f (t) be of exponential order and −∞ < a < ∞. Then L(eat f (t)) = L(f (t))|s→(s−a) . Theorem 9 (Second Shifting Rule) Let f (t) and g(t) be of exponential order and assume a ≥ 0. Then (a) (b) L(f (t − a)H(t − a)) = e−as L(f (t)), L(g(t)H(t − a)) = e−as L(g(t + a)).

Theorem 10 (Periodic Function Rule) Let f (t) be of exponential order and satisfy f (t + P ) = f (t). Then L(f (t)) =
P 0

f (t)e−st dt . 1 − e−P s

Theorem 11 (Convolution Rule) Let f (t) and g(t) be of exponential order. Then t L(f (t))L(g(t)) = L
0

f (x)g(t − x)dx .

276 Proof of Theorem 4 (linearity):
LHS = L(f (t) + g(t)) = =

Laplace Transform

Left side of the identity in (a). Direct transform. Calculus integral rule. Equals RHS; identity (a) verified. Left side of the identity in (b). Direct transform. Calculus integral rule. Equals RHS; identity (b) verified.

∞ (f (t) + g(t))e−st dt 0 ∞ ∞ f (t)e−st dt + 0 g(t)e−st dt 0

LHS = L(cf (t)) = =

= L(f (t)) + L(g(t))
∞ cf (t)e−st dt 0 ∞ c 0 f (t)e−st dt

= cL(f (t))

Proof of Theorem 5 (t-derivative rule): Already L(f (t)) exists, because f is of exponential order and continuous. On an interval [a, b] where f is continuous, integration by parts using u = e−st , dv = f (t)dt gives b a

f (t)e−st dt = f (t)e−st |t=a −

t=b

b a

f (t)(−s)e−st dt b a

= −f (a)e−sa + f (b)e−sb + s

f (t)e−st dt.

On any interval [0, N ], there are finitely many intervals [a, b] on each of which f is continuous. Add the above equality across these finitely many intervals [a, b]. The boundary values on adjacent intervals match and the integrals add to give
N 0

f (t)e−st dt = −f (0)e0 + f (N )e−sN + s

N

f (t)e−st dt.
0

Take the limit across this equality as N → ∞. Then the right side has limit −f (0) + sL(f (t)), because of the existence of L(f (t)) and limt→∞ f (t)e−st = 0 for large s. Therefore, the left side has a limit, and by definition L(f (t)) exists and L(f (t)) = −f (0) + sL(f (t)).

Proof of Theorem 6 (t-Integral rule): Let f (t) = exponential order and continuous. The details: L( t 0

t 0

g(x)dx. Then f is of

g(x)dx) = L(f (t)) 1 = L(f (t)) s 1 = L(g(t)) s

By definition. Because f (0) = 0 implies L(f (t)) = sL(f (t)). Because f = g by the Fundamental theorem of calculus.

Proof of Theorem 7 (s-differentiation): We prove the equivalent relation L((−t)f (t)) = (d/ds)L(f (t)). If f is of exponential order, then so is (−t)f (t), therefore L((−t)f (t)) exists. It remains to show the s-derivative exists and satisfies the given equality.
The proof below is based in part upon the calculus inequality (1) e−x + x − 1 ≤ x2 , x ≥ 0.

7.7 Transform Properties

277

The inequality is obtained from two applications of the mean value theorem g(b)−g(a) = g (x)(b−a), which gives e−x +x−1 = xxe−x1 with 0 ≤ x1 ≤ x ≤ x.

In addition, the existence of L(t2 |f (t)|) is used to define s0 > 0 such that L(t2 |f (t)|) ≤ 1 for s > s0 . This follows from the transform existence theorem for functions of exponential order, where it is shown that the transform has limit zero at s = ∞. Consider h = 0 and the Newton quotient Q(s, h) = (F (s + h) − F (s))/h for the s-derivative of the Laplace integral. We have to show that h→0 lim |Q(s, h) − L((−t)f (t))| = 0.

This will be accomplished by proving for s > s0 and s + h > s0 the inequality |Q(s, h) − L((−t)f (t))| ≤ |h|. For h = 0,
∞

Q(s, h) − L((−t)f (t)) =

f (t)
0

e−st−ht − e−st + the−st dt. h

Assume h > 0. Due to the exponential rule eA+B = eA eB , the quotient in the integrand simplifies to give
∞

Q(s, h) − L((−t)f (t)) =

f (t)e−st
0

e−ht + th − 1 h

dt.

Inequality (1) applies with x = ht ≥ 0, giving
∞

|Q(s, h) − L((−t)f (t))| ≤ |h|

0

t2 |f (t)|e−st dt.

The right side is |h|L(t2 |f (t)|), which for s > s0 is bounded by |h|, completing the proof for h > 0. If h < 0, then a similar calculation is made to obtain
∞

|Q(s, h) − L((−t)f (t))| ≤ |h|

0

t2 |f (t)e−st−ht dt.

The right side is |h|L(t2 |f (t)|) evaluated at s + h instead of s. If s + h > s0 , then the right side is bounded by |h|, completing the proof for h < 0.

Proof of Theorem 8 (first shifting rule): The left side LHS of the equality can be written because of the exponential rule eA eB = eA+B as
∞

LHS =
0

f (t)e−(s−a)t dt.

This integral is L(f (t)) with s replaced by s − a, which is precisely the meaning of the right side RHS of the equality. Therefore, LHS = RHS.

Proof of Theorem 9 (second shifting rule): The details for (a) are
LHS = L(H(t − a)f (t − a)) =
∞ 0

H(t − a)f (t − a)e−st dt Direct transform.

278
= = =
∞ H(t − a)f (t − a)e−st dt a ∞ −s(x+a) dx 0 H(x)f (x)e −sa ∞ −sx e dx 0 f (x)e −sa

Laplace Transform
Because a ≥ 0 and H(x) = 0 for x < 0. Change variables x = t − a, dx = dt. Direct transform. Use H(x) = 1 for x ≥ 0.

=e

= RHS

L(f (t))

Identity (a) verified.

In the details for (b), let f (t) = g(t + a), then LHS = L(H(t − a)g(t)) = e−sa L(f (t))

= L(H(t − a)f (t − a)) = e−sa L(g(t + a))

Use f (t − a) = g(t − a + a) = g(t). Apply (a). Because f (t) = g(t + a). Identity (b) verified.

= RHS

Proof of Theorem 10 (periodic function rule):
LHS = L(f (t)) =
∞ 0

f (t)e−st dt

Direct transform. Additivity of the integral. Change variables t = x + nP . Because f is P -periodic and eA eB = eA+B . Common factor in summation. Define r = e−P s . Sum the geometric series. Substitute r = e−P s . Periodic function identity verified.

= = = = = =

nP +P ∞ f (t)e−st dt n=0 nP P ∞ −sx−nP s dx n=0 0 f (x + nP )e P ∞ −nP s −sx dx n=0 e 0 f (x)e P 0 P 0 P 0

f (x)e−sx dx f (x)e−sx dx

∞ n n=0 r

f (x)e−sx dx 1 − e−P s = RHS

1 1−r

Left unmentioned here is the convergence of the infinite series on line 3 of the proof, which follows from f of exponential order.

Proof of Theorem 11 (convolution rule): The details use Fubini’s integration interchange theorem for a planar unbounded region, and therefore this proof involves advanced calculus methods that may be outside the background of the reader. Modern calculus texts contain a less general version of Fubini’s theorem for finite regions, usually referenced as iterated integrals. The unbounded planar region is written in two ways: D = {(r, t) : t ≤ r < ∞, 0 ≤ t < ∞},

D = {(r, t) : 0 ≤ r < ∞, 0 ≤ r ≤ t}. Readers should pause here and verify that D = D.

7.7 Transform Properties

279

The change of variable r = x + t, dr = dx is applied for fixed t ≥ 0 to obtain the identity (2) e−st
∞ 0

g(x)e−sx dx = =

∞ 0 ∞ t

g(x)e−sx−st dx g(r − t)e−rs dr.

The left side of the convolution identity is expanded as follows: LHS = L(f (t))L(g(t)) = = = = = Then RHS = L = = = t 0 ∞ 0 ∞ 0

f (t)e−st dt
∞ t

∞ 0

g(x)e−sx dx Direct transform. Apply identity (2). Fubini’s theorem applied. Descriptions D and D are the same. Fubini’s theorem applied.

f (t)

−rs drdt D f (t)g(r − t)e D f (t)g(r − t)e ∞ r f (t)g(r − 0 0 −rs

g(r − t)e−rs drdt drdt

t)dte

−rs

dr

f (u)g(t − u)du Direct transform. Change variable names u ↔ t. Change variable names r ↔ t.

= LHS

∞ t −st dt 0 f (u)g(t − u)due 0 ∞ r −sr dr 0 0 f (u)g(r − u)due ∞ r −sr f (t)g(r − t)dt e dr 0 0

Convolution identity verified.

280

Laplace Transform

7.8 More on the Laplace Transform
Model conversion and engineering. A differential equation model for a physical system can be subjected to the Laplace transform in order to produce an algebraic model in the transform variable s. Lerch’s theorem says that both models are equivalent, that is, the solution of one model gives the solution to the other model. In electrical and computer engineering it is commonplace to deal only with the Laplace algebraic model. Engineers are in fact capable of having hour-long modeling conversations, during which differential equations are never referenced! Terminology for such modeling is necessarily specialized, which gives rise to new contextual meanings to the terms input and output. For example, an RLC-circuit would be discussed with input F (s) = s2 ω , + ω2

and the listener must know that this expression is the Laplace transform of the t-expression sin ωt. Hence the RLC-circuit is driven by a sinusoindal input of natural frequency ω. During the modeling discourse, it could be that the output is X(s) = 10ω 1 . + 2 s + 1 s + ω2

Lerch’s equivalence says that X(s) is the Laplace transform of e−t + 10 sin ωt, but that is extra work, if all that is needed from the model is a statement about the transient and steady-state responses to the input.