Laplace transform "ℒ" redirects here. For the Lagrangian, see Lagrangian century by Abel, Lerch, Heaviside, and Bromwich. mechanics. From 1744, Leonhard Euler investigated integrals of the form In mathematics the Laplace transform is an integral transform named after its discoverer Pierre-Simon ∫ ∫ Laplace (/ləˈplɑːs/). It takes a function of a positive real z = X(x)eax dx and z = X(x)xA dx variable t (often time) to a function of a complex variable s (frequency). as solutions of differential equations but did not pursue [2] The Laplace transform is very similar to the Fourier the matter very far. transform. While the Fourier transform of a function is a complex function of a real variable (frequency), the Laplace transform of a function is a complex function of a complex variable. Laplace transforms are usually restricted to functions of t with t > 0. A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable s. Unlike the Fourier transform, the Laplace transform of a distribution is generally a well-behaved function. Also techniques of complex variables can be used directly to study Laplace transforms. As a holomorphic function, the Laplace transform has a power series representation. This power series expresses a function as a linear superposition of moments of the function. This perspective has applications in probability theory.
Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form ∫
X(x)e−ax ax dx,
which some modern historians have interpreted within modern Laplace transform theory.[3][4]
These types of integrals seem first to have attracted Laplace’s attention in 1782 where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.[5] However, in 1785, Laplace took the critical step forward when, rather than just looking for a solution in the form of an integral, he started to apply the The Laplace transform is invertible on a large class of transforms in the sense that was later to become popular. functions. The inverse Laplace transform takes a func- He used an integral of the form tion of a complex variable s (often frequency) and yields a function of a real variable t (time). Given a simple mathe- ∫ matical or functional description of an input or output to xs ϕ(x) dx, a system, the Laplace transform provides an alternative functional description that often simplifies the process of akin to a Mellin transform, to transform the whole of analyzing the behavior of the system, or in synthesizing a a difference equation, in order to look for solutions of new system based on a set of specifications.[1] So, for ex- the transformed equation. He then went on to apply the ample, Laplace transformation from the time domain to Laplace transform in the same way and started to derive the frequency domain transforms differential equations some of its properties, beginning to appreciate its poteninto algebraic equations and convolution into multiplica- tial power.[6] tion. It has many applications in the sciences and techLaplace also recognised that Joseph Fourier's method of nology. Fourier series for solving the diffusion equation could only apply to a limited region of space because those solutions were periodic. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in 1 History space.[7] The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform (now called the z-transform) in his work on 2 Formal definition probability theory. The current widespread use of the transform (mainly in engineering) came about soon af- The Laplace transform is a frequency-domain approach ter World War II although it had been used in the 19th for continuous time signals irrespective of whether the 1
2
2
FORMAL DEFINITION
system is stable or unstable. The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the [ −sX ] . function F(s), which is a unilateral transform defined by L{f }(s) = E e ∫
∞
F (s) =
e−st f (t) dt
0
where s is a complex number frequency parameter s = σ + iω , with real numbers σ and ω. Other notations for the Laplace transform include L{f} or alternatively L{f(t)} instead of F.
By abuse of language, this is referred to as the Laplace transform of the random variable X itself. Replacing s by −t gives the moment generating function of X. The Laplace transform has applications throughout probability theory, including first passage times of stochastic processes such as Markov chains, and renewal theory. Of particular use is the ability to recover the cumulative distribution function of a continuous random variable X by means of the Laplace transform as follows[9]
The meaning of the integral depends on types of func{ } } { tions of interest. A necessary condition for existence of [ −sX ] −1 1 −1 1 F (x) = L E e (x) = L L{f }(s) (x). X the integral is that f must be locally integrable on [0, ∞). s s For locally integrable functions that decay at infinity or are of exponential type, the integral can be understood to be a (proper) Lebesgue integral. However, for many ap- 2.2 Bilateral Laplace transform plications it is necessary to regard it to be a conditionally convergent improper integral at ∞. Still more generally, Main article: Two-sided Laplace transform the integral can be understood in a weak sense, and this is dealt with below. When one says “the Laplace transform” without qualifiOne can define the Laplace transform of a finite Borel cation, the unilateral or one-sided transform is normally measure μ by the Lebesgue integral[8] intended. The Laplace transform can be alternatively defined as the bilateral Laplace transform or two-sided Laplace transform by extending the limits of integration ∫ to be the entire real axis. If that is done the common uniL{µ}(s) = e−st dµ(t). [0,∞) lateral transform simply becomes a special case of the biAn important special case is where μ is a probability mea- lateral transform where the definition of the function besure or, even more specifically, the Dirac delta function. ing transformed is multiplied by the Heaviside step funcIn operational calculus, the Laplace transform of a mea- tion. sure is often treated as though the measure came from a The bilateral Laplace transform is defined as follows, probability density function f. In that case, to avoid potential confusion, one often writes ∫ ∞ B{f }(s) = e−st f (t) dt. ∫ ∞ −∞ L{f }(s) = e−st f (t) dt, 0−
where the lower limit of 0− is shorthand notation for ∫ ε↓0
Main article: Inverse Laplace transform
∞
lim
. −ε
This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform.
2.1
2.3 Inverse Laplace transform
Probability theory
In pure and applied probability, the Laplace transform is defined as an expected value. If X is a random variable with probability density function f, then the Laplace transform of f is given by the expectation
Two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero. This means that, on the range of the transform, there is an inverse transform. In fact, besides integrable functions, the Laplace transform is a one-to-one mapping from one function space into another in many other function spaces as well, although there is usually no easy characterization of the range. Typical function spaces in which this is true include the spaces of bounded continuous functions, the space L∞ (0, ∞), or more generally tempered functions (that is, functions of at worst polynomial growth) on (0, ∞). The Laplace transform is also defined and injective for suitable spaces of tempered distributions.
3 In these cases, the image of the Laplace transform lives in a space of analytic functions in the region of convergence. The inverse Laplace transform is given by the following complex integral, which is known by various names (the Bromwich integral, the Fourier–Mellin integral, and Mellin’s inverse formula):
f (t) = L−1 {F }(t) =
1 lim 2πi T →∞
∫
γ+iT
est F (s) ds, γ−iT
sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence. Similarly, the set of values for which F(s) converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the region of convergence (ROC). If the Laplace transform converges (conditionally) at s = s0 , then it automatically converges for all s with Re(s) > Re(s0 ). Therefore, the region of convergence is a half-plane of the form Re(s) > a, possibly including some points of the boundary line Re(s) = a.
where γ is a real number so that the contour path of integration is in the region of convergence of F(s). An alter- In the region of convergence Re(s) > Re(s0 ), the Laplace native formula for the inverse Laplace transform is given transform of f can be expressed by integrating by parts by Post’s inversion formula. The limit here is interpreted as the integral in the weak-* topology. ∫ ∞ ∫ u In practice, it is typically more convenient to decompose e−(s−s0 )t β(t) dt, β(u) = e−s0 t f (t) dt. a Laplace transform into known transforms of functions F (s) = (s−s0 ) 0 0 obtained from a table, and construct the inverse by inspection. That is, in the region of convergence F(s) can effectively
3
Region of convergence
be expressed as the absolutely convergent Laplace transform of some other function. In particular, it is analytic.
There are several Paley–Wiener theorems concerning the relationship between the decay properties of f and the If f is a locally integrable function (or more generally properties of the Laplace transform within the region of a Borel measure locally of bounded variation), then the convergence. Laplace transform F(s) of f converges provided that the In engineering applications, a function corresponding to limit a linear time-invariant (LTI) system is stable if every bounded input produces a bounded output. This is equiv∫ R alent to the absolute convergence of the Laplace trans−st form of the impulse response function in the region Re(s) lim f (t)e dt R→∞ 0 ≥ 0. As a result, LTI systems are stable provided the poles of the Laplace transform of the impulse response function exists. have negative real part. The Laplace transform converges absolutely if the inteThis ROC is used in knowing about the causality and stagral bility of a system. ∫ 0
∞
f (t)e−st dt
4 Properties and theorems
exists (as a proper Lebesgue integral). The Laplace transThe Laplace transform has a number of properties that form is usually understood as conditionally convergent, meaning that it converges in the former instead of the lat- make it useful for analyzing linear dynamical systems. The most significant advantage is that differentiation and ter sense. integration become multiplication and division, respecThe set of values for which F(s) converges absolutely is tively, by s (similarly to logarithms changing multiplicaeither of the form Re(s) > a or else Re(s) ≥ a, where a tion of numbers to addition of their logarithms). is an extended real constant, −∞ ≤ a ≤ ∞. (This follows from the dominated convergence theorem.) The constant Because of this property, the Laplace variable s is also in the L domain: either a is known as the abscissa of absolute convergence, and known as operator variable −1 derivative operator or (for s ) integration operator. The [10] depends on the growth behavior of f(t). Analogously, integral equations and differential equatransform turns the two-sided transform converges absolutely in a strip tions to polynomial equations, which are much easier to of the form a < Re(s) < b, and possibly including the solve. Once solved, use of the inverse Laplace transform [11] The subset of values of lines Re(s) = a or Re(s) = b. reverts to the time domain. s for which the Laplace transform converges absolutely is called the region of absolute convergence or the do- Given the functions f(t) and g(t), and their respective main of absolute convergence. In the two-sided case, it is Laplace transforms F(s) and G(s),
4
4 PROPERTIES AND THEOREMS
f (t) = L−1 {F (s)}, g(t) = L−1 {G(s)},
In other words, the Laplace transform is a continuous analog of a power series in which the discrete parameter n is replaced by the continuous parameter t, and x is replaced by e−s .
The following Table is a list of properties of unilateral Laplace transform:[12] 4.2 • Initial value theorem: f (0+ ) = lim sF (s). s→∞
Relation to moments
Main article: Moment generating function The quantities
• Final value theorem: f (∞) = lims→0 sF (s) , if all poles of sF(s) are in the left half-plane. The final value theorem is useful because it gives the long-term behaviour without having to perform partial fraction decompositions or other difficult algebra. If F(s) has a pole in the right-hand plane or poles on the imaginary axis (e.g., if f (t) = et or f (t) = sin(t) ), the behaviour of this formula is undefined.
4.1
Relation to power series
∫ µn =
∞
tn f (t) dt
0
are the moments of the function f. If the first n moments of f converge absolutely, then by repeated differentiation under the integral,
(−1)n (Lf )(n) (0) = µn . This is of special significance in probability theory, where the moments of a random variable X are given by the expectation values µn = E[X n ] . Then, the relation holds
The Laplace transform can be viewed as a continuous n analogue of a power series. If a(n) is a discrete function µ = (−1)n d E [e−sX ] (0). n dsn of a positive integer n, then the power series associated to a(n) is the series ∞ ∑
4.3 Proof of the Laplace transform of a function’s derivative a(n)xn
It is often convenient to use the differentiation property of the Laplace transform to find the transform of a function’s where x is a real variable (see Z transform). Replacing derivative. This can be derived from the basic expression summation over n with integration over t, a continuous for a Laplace transform as follows: version of the power series becomes n=0
∫
∫
∞
f (t)x dt 0
where the discrete function a(n) is replaced by the continuous one f(t). (See Mellin transform below.) Changing the base of the power from x to e gives ∫
∞
( )t f (t) eln x dt
∞
e−st f (t) dt [ ]∞ ∫ ∞ −st f (t)e−st e = − f ′ (t) dt −s 0− −s 0− [ ] f (0− ) 1 = − + L {f ′ (t)} , −s s
L {f (t)} =
t
0−
yielding
0
For this to converge for, say, all bounded functions f, it is L {f ′ (t)} = s · L {f (t)} − f (0− ), necessary to require that ln x < 0. Making the substitution and in the bilateral case, −s = ln x gives just the Laplace transform: ∫
∞
f (t)e 0
−st
dt
′
L {f (t)} = s
∫
∞
−∞
e−st f (t) dt = s · L{f (t)}.
parts) (by
4.5
Relationship to other transforms
5
The general result
then the Laplace–Stieltjes transform of g and the Laplace transform of f coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of the Stieltjes mea{ } (n) n n−1 − (n−1) − sure associated to g. So in practice, the only distincL f (t) = s ·L {f (t)}−s f (0 )−· · ·−f (0 ), tion between the two transforms is that the Laplace transwhere f (n) denotes the nth derivative of f, can then be form is thought of as operating on the density function of the measure, whereas the Laplace–Stieltjes transform established with an inductive argument. is thought of as operating on its cumulative distribution function.[14]
4.4
Evaluating improper integrals
Let L {f (t)} = F (s) , then (see the table above) { L
f (t) t
}
∫
∞
=
F (p) dp,
fˆ(ω) = F{f (t)} ∞
0
f (t) −st e dt = t
∫
∞
0
f (t) dt = t
∫
F (p) dp. s
∞
F (p) dp. 0
provided that the interchange of limits can be justified. Even when the interchange cannot be justified the calculation can be suggestive. For example, proceeding formally one has ∫
∞
0
= L{f (t)}|s=iω = F (s)|s=iω ∫ ∞ = e−iωt f (t) dt .
∞
Letting s → 0, gives one the identity ∫
The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument s = iω or s = 2πfi,[15]
s
or ∫
4.5.2 Fourier transform
−∞
This definition of the Fourier transform requires a prefactor of 1/2 π on the reverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine the frequency spectrum of a signal or dynamical system. The above relation is valid as stated if and only if the region of convergence (ROC) of F(s) contains the imaginary axis, σ = 0.
For example, the function f(t) = cos(ω0 t) has a Laplace transform F(s) = s/(s2 + ω0 2 ) whose ROC is Re(s) > 0. As)s = iω is a pole of F(s), ∞ substituting s = iω in F(s) ∫ ∞( 1 p p does not yield 1 the p2Fourier + a2 transform of f(t)u(t), which is (cos(at) − cos(bt)) dt = − 2 dp = ln 2 = ln b−ln a. 2 Dirac t p2 + a2 p + bproportional 2 to the p + b2 0delta-function δ(ω − ω0 ). 0
The validity of this identity can be proved by other means. However, a relation of the form It is an example of a Frullani integral. Another example is Dirichlet integral.
lim F (σ + iω) = fˆ(ω)
σ→0+
holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a weak limit of measures (see vague topol4.5.1 Laplace–Stieltjes transform ogy). General conditions relating the limit of the Laplace transform of a function on the boundary to the Fourier The (unilateral) Laplace–Stieltjes transform of a function transform take the form of Paley-Wiener theorems. g : R → R is defined by the Lebesgue–Stieltjes integral
4.5
Relationship to other transforms
{L∗ g}(s) =
∫
4.5.3 Mellin transform ∞
e−st dg(t).
0
The function g is assumed to be of bounded variation. If g is the antiderivative of f: ∫ g(x) =
If in the Mellin transform ∫
x
f (t) dt 0
The Mellin transform and its inverse are related to the two-sided Laplace transform by a simple change of variables.
G(s) = M{g(θ)} = 0
∞
θs g(θ)
dθ θ
6
5 TABLE OF SELECTED LAPLACE TRANSFORMS
we set θ = e−t we get a two-sided Laplace transform. 4.5.4
Z-transform
Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal,
The unilateral or one-sided Z-transform is simply the X (s) = X(z) sT . q Laplace transform of an ideally sampled signal with the z=e substitution of The similarity between the Z and Laplace transforms is expanded upon in the theory of time scale calculus. def
z = esT , where T = 1/fs is the sampling period (in units of time 4.5.5 Borel transform e.g., seconds) and fs is the sampling rate (in samples per The integral form of the Borel transform second or hertz). Let
∫ def
∆T (t) =
∞
F (s) =
∞ ∑
f (z)e−sz dz
0
δ(t − nT )
is a special case of the Laplace transform for f an entire function of exponential type, meaning that
n=0
be a sampling impulse train (also called a Dirac comb) and |f (z)| ≤ AeB|z| def
xq (t) = x(t)∆T (t) = x(t)
∞ ∑
δ(t − nT )
n=0
=
∞ ∑
x(nT )δ(t − nT ) =
n=0
∞ ∑
x[n]δ(t − nT )
n=0
be the sampled representation of the continuous-time x(t)
for some constants A and B. The generalized Borel transform allows a different weighting function to be used, rather than the exponential function, to transform functions not of exponential type. Nachbin’s theorem gives necessary and sufficient conditions for the Borel transform to be well defined. 4.5.6 Fundamental relationships
def
x[n] = x(nT ) . The Laplace transform of the sampled signal xq₍t₎ is ∫ Xq (s) = = = =
∞
xq (t)e 0− ∫ ∞∑ ∞
−st
dt
x[n]δ(t − nT )e−st dt
0− n=0 ∫ ∞ ∞ ∑
x[n]
n=0 ∞ ∑
0−
δ(t − nT )e−st dt
x[n]e−nsT .
Since an ordinary Laplace transform can be written as a special case of a two-sided transform, and since the twosided transform can be written as the sum of two onesided transforms, the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject. However, a different point of view and different characteristic problems are associated with each of these four major integral transforms.
5 Table of selected Laplace transforms
n=0
The following table provides Laplace transforms for This is the precise definition of the unilateral Z-transform many common functions of a single variable.[16][17] For of the discrete function x[n] definitions and explanations, see the Explanatory Notes at the end of the table. X(z) =
∞ ∑
x[n]z −n
n=0
with the substitution of z → esT .
Because the Laplace transform is a linear operator, • The Laplace transform of a sum is the sum of Laplace transforms of each term.
7
L{f (t) + g(t)} = L{f (t)} + L{g(t)}
initial conditions on the circuit elements. For example, if a capacitor has an initial voltage across it, or if the inductor has an initial current through it, the sources inserted in the s-domain account for that.
• The Laplace transform of a multiple of a function The equivalents for current and voltage sources are simply is that multiple times the Laplace transformation of derived from the transformations in the table above. that function.
L{af (t)} = aL{f (t)}
7 Examples: How to apply the properties and theorems
Using this linearity, and various trigonometric, hyperbolic, and complex number (etc.) properties and/or identities, some Laplace transforms can be obtained from others quicker than by using the definition directly.
The Laplace transform is used frequently in engineering and physics; the output of a linear time-invariant system can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; The unilateral Laplace transform takes as input a func- the latter being easier to solve because of its algebraic tion whose time domain is the non-negative reals, which form. For more information, see control theory. is why all of the time domain functions in the table below The Laplace transform can also be used to solve difare multiples of the Heaviside step function, u(t). ferential equations and is used extensively in electrical engineering. The Laplace transform reduces a linear The entries of the table that involve a time delay τ are redifferential equation to an algebraic equation, which can quired to be causal (meaning that τ > 0). A causal system then be solved by the formal rules of algebra. The origis a system where the impulse response h(t) is zero for all inal differential equation can then be solved by applying time t prior to t = 0. In general, the region of convergence the inverse Laplace transform. The English electrical enfor causal systems is not the same as that of anticausal sysgineer Oliver Heaviside first proposed a similar scheme, tems. although without using the Laplace transform; and the resulting operational calculus is credited as the Heaviside calculus.
6
s-domain equivalent circuits and impedances
7.1 Example 1: equation The Laplace transform is often used in circuit analysis,
Solving a differential
and simple conversions to the s-domain of circuit eleIn nuclear physics, the following fundamental relationments can be made. Circuit elements can be transformed ship governs radioactive decay: the number of radioacinto impedances, very similar to phasor impedances. tive atoms N in a sample of a radioactive isotope decays Here is a summary of equivalents: at a rate proportional to N. This leads to the first order linear differential equation
dN = −λN, dt where λ is the decay constant. The Laplace transform can be used to solve this equation. Rearranging the equation to one side, we have
dN + λN = 0. dt s-domain equivalent circuits
Next, we take the Laplace transform of both sides of the equation:
) Note that the resistor is exactly the same in the time do- ( ˜ (s) − N0 + λN ˜ (s) = 0, sN main and the s-domain. The sources are put in if there are
8
7
EXAMPLES: HOW TO APPLY THE PROPERTIES AND THEOREMS
where
Solving for V(s) we have
˜ (s) = L{N (t)} N
V (s) =
and
The definition of the complex impedance Z (in ohms) is the ratio of the complex voltage V divided by the complex current I while holding the initial state V 0 at zero:
N0 = N (0).
V (s) . Z(s) = I(s) V0 =0
Solving, we find
˜ (s) = N0 . N s+λ
Using this definition and the previous equation, we find:
Finally, we take the inverse Laplace transform to find the general solution
−1
N (t) = L
I(s) V0 + . sC s
˜ (s)} = L−1 {N
{
N0 s+λ
}
Z(s) =
1 , sC
which is the correct expression for the complex impedance of a capacitor.
= N0 e−λt , which is indeed the correct form for radioactive decay.
7.2
7.3 Example 3: Method of partial fraction expansion
Example 2: Deriving the complex Consider a linear time-invariant system with transfer function impedance for a capacitor
In the theory of electrical circuits, the current flow in a 1 capacitor is proportional to the capacitance and rate of H(s) = . (s + α)(s + β) change in the electrical potential (in SI units). Symbolically, this is expressed by the differential equation The impulse response is simply the inverse Laplace transform of this transfer function: dv i=C , dt h(t) = L−1 {H(s)}. where C is the capacitance (in farads) of the capacitor, i = i(t) is the electric current (in amperes) through the To evaluate this inverse transform, we begin by expanding capacitor as a function of time, and v = v(t) is the voltage H(s) using the method of partial fraction expansion, (in volts) across the terminals of the capacitor, also as a function of time. Taking the Laplace transform of this equation, we obtain
P R 1 = + . (s + α)(s + β) s+α s+β
I(s) = C(sV (s) − V0 ),
The unknown constants P and R are the residues located at the corresponding poles of the transfer function. Each residue represents the relative contribution of that singularity to the transfer function’s overall shape.
where I(s) = L{i(t)}, V (s) = L{v(t)}, and
V0 = v(t)|t=0 .
By the residue theorem, the inverse Laplace transform depends only upon the poles and their residues. To find the residue P, we multiply both sides of the equation by s + α to get
R(s + α) 1 =P+ . s+β s+β
7.6
Example 5: Phase delay
9
Then by letting s = −α, the contribution from R vanishes and all that is left is X(s) = 1 1 P = = . s + β s=−α β−α
s+β , (s + α)2 + ω 2
we find the inverse transform by first adding and subtracting the same constant α to the numerator:
Similarly, the residue R is given by
R=
X(s) =
1 1 = . s + α s=−β α−β
s+α β−α + . 2 2 (s + α) + ω (s + α)2 + ω 2
By the shift-in-frequency property, we have
Note that x(t) = e−αt L−1 R=
−1 = −P β−α
= e−αt L−1
and so the substitution of R and P into the expanded expression for H(s) gives ( H(s) =
1 β−α
) ( ·
1 1 − s+α s+β
) .
Finally, using the linearity property and the known transform for exponential decay (see Item #3 in the Table of Laplace Transforms, above), we can take the inverse Laplace transform of H(s) to obtain
h(t) = L−1 {H(s)} =
) 1 ( −αt e − e−βt , β−α
which is the impulse response of the system.
7.4
{ {
β−α s + 2 s2 + ω 2 s + ω2 s + s2 + ω 2
[ { = e−αt L−1
s s2 + ω 2
(
β−α ω
}
( +
} )(
β−α ω
ω s2 + ω 2
)
L−1
{
)}
ω s2 + ω 2
Finally, using the Laplace transforms for sine and cosine (see the table, above), we have [ ( ) ] β−α x(t) = e cos (ωt)u(t) + sin (ωt)u(t) . ω [ ( ) ] β−α −αt x(t) = e cos (ωt) + sin (ωt) u(t). ω −αt
7.6 Example 5: Phase delay Starting with the Laplace transform,
Example 3.2: Convolution
s sin(ϕ) + ω cos(ϕ) X(s) = The same result can be achieved using the convolution s2 + ω 2 property as if the system is a series of filters with transfer functions of 1/(s + a) and 1/(s + b). That is, the inverse we find the inverse by first rearranging terms in the fraction: of
H(s) =
1 1 1 = · (s + a)(s + b) s+a s+b
is
L−1
7.5
{
s sin(ϕ) ω cos(ϕ) + 2 s2 + ω 2 s + ω2 ( ) ( ) s ω = sin(ϕ) 2 + cos(ϕ) 2 . s + ω2 s + ω2
X(s) =
We are now able to take the inverse Laplace transform of } { } ∫ t our terms: e−at − e−bt 1 1 ∗L−1 = e−at ∗e−bt = e−ax e−b(t−x) dx = . s+a s+b b−a 0 { } { } s ω −1 x(t) = sin(ϕ)L−1 + cos(ϕ)L Example 4: Mixing sines, cosines, and s2 + ω 2 s2 + ω 2
exponentials Starting with the Laplace transform
= sin(ϕ) cos(ωt) + sin(ωt) cos(ϕ). This is just the sine of the sum of the arguments, yielding:
}] .
10
9 NOTES
9 Notes x(t) = sin(ωt + ϕ). We can apply similar logic to find that
L−1
7.7
{
s cos ϕ − ω sin ϕ s2 + ω 2
[1] Korn & Korn 1967, §8.1 [2] Euler 1744, Euler 1753, Euler 1769
} = cos (ωt + ϕ).
Example 6: Determining structure of astronomical object from spectrum
The wide and general applicability of the Laplace transform and its inverse is illustrated by an application in astronomy which provides some information on the spatial distribution of matter of an astronomical source of radiofrequency thermal radiation too distant to resolve as more than a point, given its flux density spectrum, rather than relating the time domain with the spectrum (frequency domain). Assuming certain properties of the object, e.g. spherical shape and constant temperature, calculations based on carrying out an inverse Laplace transformation on the spectrum of the object can produce the only possible model of the distribution of matter in it (density as a function of distance from the center) consistent with the spectrum.[20] When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement.
[3] Lagrange 1773 [4] Grattan-Guinness 1997, p. 260 [5] Grattan-Guinness 1997, p. 261 [6] Grattan-Guinness 1997, pp. 261–262 [7] Grattan-Guinness 1997, pp. 262–266 [8] Feller 1971, §XIII.1 [9] The cumulative distribution function is the integral of the probability density function. [10] Widder 1941, Chapter II, §1 [11] Widder 1941, Chapter VI, §2 [12] Korn & Korn 1967, pp. 226–227 [13] Bracewell 2000, Table 14.1, p. 385 [14] Feller 1971, p. 432 [15] Takacs 1953, p. 93
8
See also • Analog signal processing • Bernstein’s theorem on monotone functions • Continuous-repayment mortgage • Fourier transform • Hamburger moment problem • Hardy–Littlewood tauberian theorem • Moment-generating function • N-transform • Pierre-Simon Laplace • Post’s inversion formula • Signal-flow graph • Laplace–Carson transform • Symbolic integration • Transfer function • Z-transform (discrete equivalent of the Laplace transform)
[16] Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010), Mathematical methods for physics and engineering (3rd ed.), Cambridge University Press, p. 455, ISBN 978-0-52186153-3 [17] Distefano, J. J.; Stubberud, A. R.; Williams, I. J. (1995), Feedback systems and control, Schaum’s outlines (2nd ed.), McGraw-Hill, p. 78, ISBN 0-07-017052-5 [18] Lipschutz, S.; Spiegel, M. R.; Liu, J. (2009), Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series (3rd ed.), McGraw-Hill, p. 183, ISBN 978-0-07154855-7 – provides the case for real q. [19] http://mathworld.wolfram.com/LaplaceTransform.html – Wolfram Mathword provides case for complex q [20] Salem, M.; Seaton, M. J. (1974), “I. Continuum spectra and brightness contours”, Monthly Notices of the Royal Astronomical Society 167: 493–510, Bibcode:1974MNRAS.167..493S, doi:10.1093/mnras/167.3.493, and Salem, M. (1974), “II. Three-dimensional models”, Monthly Notices of the Royal Astronomical Society 167: 511–516, Bibcode:1974MNRAS.167..511S, doi:10.1093/mnras/167.3.511
11
10 10.1
References Modern
• Bracewell, Ronald N. (1978), The Fourier Transform and its Applications (2nd ed.), McGraw-Hill Kogakusha, ISBN 0-07-007013-X • Bracewell, R. N. (2000), The Fourier Transform and Its Applications (3rd ed.), Boston: McGraw-Hill, ISBN 0-07-116043-4 • Feller, William (1971), An introduction to probability theory and its applications. Vol. II., Second edition, New York: John Wiley & Sons, MR 0270403 • Korn, G. A.; Korn, T. M. (1967), Mathematical Handbook for Scientists and Engineers (2nd ed.), McGraw-Hill Companies, ISBN 0-07-035370-0 • Widder, David Vernon (1941), The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, MR 0005923 • Williams, J. (1973), Laplace Transforms, Problem Solvers, George Allen & Unwin, ISBN 0-04512021-8 • Takacs, J. (1953), “Fourier amplitudok meghatarozasa operatorszamitassal”, Magyar Hiradastechnika (in Hungarian) IV (7–8): 93–96
10.2
Historical
• Euler, L. (1744), “De constructione aequationum” [The Construction of Equations], Opera omnia, 1st series (in Latin) 22: 150–161 • Euler, L. (1753), “Methodus aequationes differentiales” [A Method for Solving Differential Equations], Opera omnia, 1st series (in Latin) 22: 181– 213 • Euler, L. (1992) [1769], “Institutiones calculi integralis, Volume 2” [Institutions of Integral Calculus], Opera omnia, 1st series (in Latin) (Basel: Birkhäuser) 12, ISBN 978-3764314743, Chapters 3–5
11 Further reading • Arendt, Wolfgang; Batty, Charles J.K.; Hieber, Matthias; Neubrander, Frank (2002), VectorValued Laplace Transforms and Cauchy Problems, Birkhäuser Basel, ISBN 3-7643-6549-8. • Davies, Brian (2002), Integral transforms and their applications (Third ed.), New York: Springer, ISBN 0-387-95314-0 • Deakin, M. A. B. (1981), “The development of the Laplace transform”, Archive for History of Exact Sciences 25 (4): 343–390, doi:10.1007/BF01395660 • Deakin, M. A. B. (1982), “The development of the Laplace transform”, Archive for History of Exact Sciences 26 (4): 351–381, doi:10.1007/BF00418754 • Doetsch, Gustav (1974), Introduction to the Theory and Application of the Laplace Transformation, Springer, ISBN 0-387-06407-9 • Mathews, Jon; Walker, Robert L. (1970), Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin, ISBN 0-8053-7002-1 • Polyanin, A. D.; Manzhirov, A. V. (1998), Handbook of Integral Equations, Boca Raton: CRC Press, ISBN 0-8493-2876-4 • Schwartz, Laurent (1952), “Transformation de Laplace des distributions”, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] (in French) 1952: 196–206, MR 0052555 • Siebert, William McC. (1986), Circuits, Signals, and Systems, Cambridge, Massachusetts: MIT Press, ISBN 0-262-19229-2 • Widder, David Vernon (1945), “What is the Laplace transform?", The American Mathematical Monthly (MAA) 52 (8): 419–425, doi:10.2307/2305640, ISSN 0002-9890, JSTOR 2305640, MR 0013447
12 External links
• Euler, Leonhard (1769), Institutiones calculi integralis [Institutions of Integral Calculus] (in Latin) II, Paris: Petropoli, ch. 3–5, pp. 57–153
• Hazewinkel, Michiel, ed. (2001), “Laplace transform”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Grattan-Guinness, I (1997), “Laplace’s integral solutions to partial differential equations”, in Gillispie, C. C., Pierre Simon Laplace 1749–1827: A Life in Exact Science, Princeton: Princeton University Press, ISBN 0-691-01185-0
• Online Computation of the transform or inverse transform, wims.unice.fr
• Lagrange, J. L. (1773), Mémoire sur l'utilité de la méthode, Œuvres de Lagrange 2, pp. 171–234
• Weisstein, Eric MathWorld.
• Tables of Integral Transforms at EqWorld: The World of Mathematical Equations. W.,
“Laplace
Transform”,
12 • Laplace Transform Module by John H. Mathews • Good explanations of the initial and final value theorems • Laplace Transforms at MathPages • Computational Knowledge Engine allows to easily calculate Laplace Transforms and its inverse Transform.
12
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