11.2. LAPLACE TRANSFORM 437 11.2.1-2. Inverse Laplace transform. Given the transform  f(p), the function f(x) can be found by means of the inverse Laplace transform f(x)= 1 2πi  c+i∞ c–i∞  f(p)e px dp, i 2 =–1,(11.2.1.2) where the integration path is parallel to the imaginary axis and lies to the right of all singularities of  f(p), which corresponds to c > σ 0 . The integral in inversion formula (11.2.1.2) is understood in the sense of the Cauchy principal value:  c+i∞ c–i∞  f(p)e px dp = lim ω→∞  c+iω c–iω  f(p)e px dp. In the domain x < 0, formula (11.2.1.2) gives f(x) ≡ 0. Formula (11.2.1.2) holds forcontinuous functions. If f (x)hasa(finite) jump discontinu- ity at a point x = x 0 > 0, then the left-hand side of (11.2.1.2) is equal to 1 2 [f(x 0 –0)+f(x 0 +0)] at this point (for x 0 = 0,thefirst term in the square brackets must be omitted). For brevity, we write the Laplace inversion formula (11.2.1.2) as follows: f(x)=L –1   f(p)  or f(x)=L –1   f(p), x  . There are tables of direct and inverse Laplace transforms (see Sections T3.1 and T3.2), which are handy in solving linear differential and integral equations. 11.2.2. Main Properties of the Laplace Transform. Inversion Formulas for Some Functions 11.2.2-1. Convolution theorem. Main properties of the Laplace transform. 1 ◦ .Theconvolution of two functions f(x)andg(x)isdefined as an integral of the form  x 0 f(t)g(x – t) dt, and is usually denoted by f (x) ∗ g(x)or f(x) ∗ g(x)=  x 0 f(t) g(x – t) dt. By performing substitution x – t = u, we see that the convolution is symmetric with respect to the convolved functions: f(x) ∗ g(x)=g(x) ∗f (x). The convolution theorem states that L  f(x) ∗ g(x)  = L  f(x)  L  g(x)  and is frequentlyapplied to solve Volterra equations with kernels depending on the difference of the arguments. 2 ◦ . The main properties of the correspondence between functions and their Laplace trans- forms are gathered in Table 11.1. 3 ◦ . The Laplace transforms of some functions are listed in Table 11.2; for more detailed tables, see Section T3.1 and the list of references at the end of this chapter. 438 INTEGRAL TRANSFORMS TABLE 11.1 Main properties of the Laplace transform No. Function Laplace transform Operation 1 af 1 (x)+bf 2 (x) a  f 1 (p)+b  f 2 (p) Linearity 2 f(x/a), a > 0 a  f(ap) Scaling 3 f(x – a), f(ξ) ≡ 0 for ξ < 0 e –ap  f(p) Shift of the argument 4 x n f(x); n = 1, 2, (–1) n  f (n) p (p) Differentiation of the transform 5 1 x f(x)  ∞ p  f (q) dq Integration of the transform 6 e ax f(x)  f(p – a) Shift in the complex plane 7 f  x (x) p  f(p)–f(+0) Differentiation 8 f (n) x (x) p n  f(p)– n  k=1 p n–k f (k–1) x (+0) Differentiation 9 x m f (n) x (x), m = 1, 2, (–1) m d m dp m  p n  f (p)– n  k=1 p n–k f (k–1) x (+0)  Differentiation 10 d n dx n  x m f(x)  , m ≥ n (–1) m p n d m dp m  f (p) Differentiation 11  x 0 f(t) dt  f(p) p Integration 12  x 0 f 1 (t)f 2 (x – t)dt  f 1 (p)  f 2 (p) Convolution TABLE 11.2 The Laplace transforms of some functions No. Function, f(x) Laplace transform,  f (p) Remarks 1 1 1/p 2 x n n! p n+1 n = 1, 2, 3 x a Γ(a + 1)p –a–1 a >–1 4 e –ax (p + a) –1 5 x a e –bx Γ(a + 1)(p + b) –a–1 a >–1 6 sinh(ax) a p 2 – a 2 7 cosh(ax) p p 2 – a 2 8 ln x – 1 p (ln p + C) C = 0.5772 is the Euler constant 9 sin(ax) a p 2 + a 2 10 cos(ax) p p 2 + a 2 11 erfc  a 2 √ x  1 p exp  –a √ p  a ≥ 0 12 J 0 (ax) 1  p 2 + a 2 J 0 (x) is the Bessel function 11.2. LAPLACE TRANSFORM 439 11.2.2-2. Inverse transforms of rational functions. Consider the important case in which the transform is a rational function of the form  f(p)= R(p) Q(p) ,(11.2.2.1) where Q(p)andR(p) are polynomials in the variable p and the degree of Q(p) exceeds that of R(p). Assume that the zeros of the denominator are simple, i.e., Q(p) ≡ const (p – λ 1 )(p – λ 2 ) (p – λ n ). Then the inverse transform can be determined by the formula f(x)= n  k=1 R(λ k ) Q  (λ k ) exp(λ k x), (11.2.2.2) where the primes denote the derivatives. If Q(p) has multiple zeros, i.e., Q(p) ≡ const (p – λ 1 ) s 1 (p – λ 2 ) s 2 (p – λ m ) s m , then f(x)= m  k=1 1 (s k – 1)! lim p→s k d s k –1 dp s k –1  (p – λ k ) s k  f(p)e px  .(11.2.2.3) Example 1. The transform  f(p)= b p 2 – a 2 (a, b real numbers) can be represented as the fraction (11.2.2.1) with R(p)=b and Q(p)=(p – a)(p + a). The denominator Q(p) has two simple roots, λ 1 = a and λ 2 =–a. Using formula (11.2.2.2) with n = 2 and Q  (p)=2p, we obtain the inverse transform in the form f(x)= b 2a e ax – b 2a e –ax = b a sinh(ax). Example 2. The transform  f(p)= b p 2 + a 2 (a, b real numbers) can be written as the fraction (11.2.2.1) with R(p)=b and Q(p)=(p – ia)(p + ia), i 2 =–1. The denominator Q(p) has two simple pure imaginary roots, λ 1 = ia and λ 2 =–ia. Using formula (11.2.2.2) with n = 2,wefind the inverse transform: f(x)= b 2ia e iax – b 2ia e –iax =– bi 2a  cos(ax)+i sin(ax)  + bi 2a  cos(ax)–i sin(ax)  = b a sin(ax). Example 3. The transform  f(p)=ap –n , where n is a positive integer, can be written as the fraction (11.2.2.1) with R(p)=a and Q(p)=p n .The denominator Q(p) has one root of multiplicity n, λ 1 = 0. By formula (11.2.2.3) with m = 1 and s 1 = n,we find the inverse transform: f(x)= a (n – 1)! x n–1 .  Fairly detailed tables of inverse Laplace transforms can be found in Section T3.2. 440 INTEGRAL TRANSFORMS 11.2.2-3. Inversion of functions with finitely many singular points. If the function  f(p)hasfinitely many singular points, p 1 , p 2 , , p n , and tends to zero as p →∞, then the integral in the Laplace inversion formula (11.2.1.2) may be evaluated using the residue theory by applying the Jordan lemma (see Subsection 11.1.2). In this case f(x)= n  k=1 res p=p k [  f(p)e px ]. (11.2.2.4) Formula (11.2.2.4) can be extended to the case where  f(p)hasinfinitely many singular points. In this case, f(x) is represented as an infinite series. 11.2.3. Limit Theorems. Representation of Inverse Transforms as Convergent Series and Asymptotic Expansions 11.2.3-1. Limit theorems. THEOREM 1. Let 0 ≤ x < ∞ and  f(p)=L  f(x)  be the Laplace transform of f(x) .If a limit of f(x) as x → 0 exists, then lim x→0 f(x) = lim p→∞  p  f(p)  . T HEOREM 2. If a limit of f(x) as x →∞ exists, then lim x→∞ f(x) = lim p→0  p  f(p)  . 11.2.3-2. Representation of inverse transforms as convergent series. THEOREM 1. Suppose the transform  f(p) can be expanded into series in negative powers of p ,  f(p)= ∞  n=1 a n p n , convergent for |p| > R ,where R is an arbitrary positive number; note that the transform tends to zero as |p| →∞ . Then the inverse transform can be obtained by the formula f(x)= ∞  n=1 a n (n – 1)! x n–1 , where the series on the right-hand side is convergent for all x . THEOREM 2. Suppose the transform  f(p) , |p| > R , is represented by an absolutely convergent series,  f(p)= ∞  n=0 a n p λ n ,(11.2.3.1) 11.3. MELLIN TRANSFORM 441 where {λ n } is any positive increasing sequence, 0 < λ 0 < λ 1 < ···→∞ . Then it is possible to proceed termwise from series (11.2.3.1) to the following inverse transform series: f(x)= ∞  n=0 a n Γ(λ n ) x λ n –1 ,(11.2.3.2) where Γ(λ) is the Gamma function. Series (11.2.3.2) is convergent for all real and complex values of x other than zero (if λ 0 ≥ 1 , the series is convergent for all x ). 11.2.3-3. Representation of inverse transforms as asymptotic expansions as x →∞. 1 ◦ .Letp = p 0 be a singular point of the Laplace transform  f(p) with the greatest real part (it is assumed there is only one such point). If  f(p) can be expanded near p = p 0 into an absolutely convergent series,  f(p)= ∞  n=0 c n (p – p 0 ) λ n (λ 0 < λ 1 < ···→∞)(11.2.3.3) with arbitrary λ n , then the inverse transform f(x) can be expressed in the form of the asymptotic expansion f(x) ∼ e p 0 x ∞  n=0 c n Γ(–λ n ) x –λ n –1 as x →∞.(11.2.3.4) The terms corresponding to nonnegative integer λ n must be omitted from the summation, since Γ(0)=Γ(–1)=Γ(–2)=···= ∞. 2 ◦ . If the transform  f(p) has several singular points, p 1 , , p m , with the same greatest real part, Re p 1 = ···=Rep m , then expansions of the form (11.2.3.3) should be obtained for each of these points and the resulting expressions must be added together. 11.2.3-4. Post–Widder formula. In applications, one can find f (x) if the Laplace transform  f(t) on the real semiaxis is known for t = p ≥ 0. To this end, one uses the Post–Widder formula f(x) = lim n→∞  (–1) n n!  n x  n+1  f (n) t  n x   .(11.2.3.5) Approximate inversion formulas are obtained by taking sufficiently large positive integer n in (11.2.3.5) instead of passing to the limit. 11.3. Mellin Transform 11.3.1. Mellin Transform and the Inversion Formula 11.3.1-1. Mellin transform. Suppose that a function f(x)isdefined for positive x and satisfies the conditions  1 0 |f(x)| x σ 1 –1 dx < ∞,  ∞ 1 |f(x)| x σ 2 –1 dx < ∞ for some real numbers σ 1 and σ 2 , σ 1 < σ 2 . 442 INTEGRAL TRANSFORMS The Mellin transform of f (x)isdefined by ˆ f(s)=  ∞ 0 f(x)x s–1 dx,(11.3.1.1) where s = σ + iτ is a complex variable (σ 1 < σ < σ 2 ). For brevity, we rewrite formula (11.3.1.1) as follows: ˆ f(s)=M{f (x)} or ˆ f(s)=M{f (x), s}. 11.3.1-2. Inverse Mellin transform. Given ˆ f(s), the function f(x) can be found by means of the inverse Mellin transform f(x)= 1 2πi  σ+i∞ σ–i∞ ˆ f(s)x –s ds (σ 1 < σ < σ 2 ), (11.3.1.2) where the integration path is parallel to the imaginary axis of the complex plane s and the integral is understood in the sense of the Cauchy principal value. Formula (11.3.1.2) holds forcontinuous functions. If f(x)hasa(finite) jump discontinu- ity at a point x = x 0 >0, then the left-hand side of (11.3.1.2) is equal to 1 2  f(x 0 –0)+f (x 0 +0)  at this point (for x 0 = 0,thefirst term in the square brackets must be omitted). For brevity, we rewrite formula (11.3.1.2) in the form f(x)=M –1 { ˆ f(s)} or f(x)=M –1 { ˆ f(s), x}. 11.3.2. Main Properties of the Mellin Transform. Relation Among the Mellin, Laplace, and Fourier Transforms 11.3.2-1. Main properties of the Mellin transform. 1 ◦ . The main properties of the correspondence between the functions and their Mellin transforms are gathered in Table 11.3. 2 ◦ . The integral relations  ∞ 0 f(x)g(x) dx = M –1 { ˆ f(s)ˆg(1 – s)},  ∞ 0 f(x)g  1 x  dx = M –1 { ˆ f(s)ˆg(s)} hold for fairly general assumptions about the integrability of the functions involved (see Ditkin and Prudnikov, 1965). 11.3.2-2. Relation among the Mellin, Laplace, and Fourier transforms. There are tables of direct and inverse Mellin transforms (see Sections T3.5 and T3.6 and the references listed at the end of the current chapter) that are useful in solving specifi c integral and differential equations. The Mellin transform is related to the Laplace and Fourier transforms as follows: M{f (x), s} = L{f (e x ), –s} + L{f(e –x ), s} = F{f(e x ), is}, which makes it possible to apply much more common tables of direct and inverse Laplace and Fourier transforms. 11.4. VARIOUS FORMS OF THE FOURIER TRANSFORM 443 TABLE 11.3 Main properties of the Mellin transform No. Function Mellin transform Operation 1 af 1 (x)+bf 2 (x) a ˆ f 1 (s)+b ˆ f 2 (s) Linearity 2 f(ax), a > 0 a –s ˆ f (s) Scaling 3 x a f(x) ˆ f(s + a) Shift of the argument of the transform 4 f(x 2 ) 1 2 ˆ f  1 2 s  Squared argument 5 f(1/x) ˆ f(–s) Inversion of the argument of the transform 6 x λ f  ax β  , a > 0, β ≠ 0 1 β a s+λ β ˆ f  s + λ β  Power law transform 7 f  x (x) –(s – 1) ˆ f (s – 1) Differentiation 8 xf  x (x) –s ˆ f (s) Differentiation 9 f (n) x (x) (–1) n Γ(s) Γ(s – n) ˆ f(s – n) Multiple differentiation 10  x d dx  n f(x) (–1) n s n ˆ f(s) Multiple differentiation 11 x α  ∞ 0 t β f 1 (xt)f 2 (t) dt ˆ f 1 (s + α) ˆ f 2 (1 – s – α + β) Complicated integration 12 x α  ∞ 0 t β f 1  x t  f 2 (t) dt ˆ f 1 (s + α) ˆ f 2 (s + α + β + 1) Complicated integration 11.4. Various Forms of the Fourier Transform 11.4.1. Fourier Transform and the Inverse Fourier Transform 11.4.1-1. Standard form of the Fourier transform. The Fourier transform is defined as follows:  f(u)= 1 √ 2π  ∞ –∞ f(x) e –iux dx.(11.4.1.1) For brevity, we rewrite formula (11.4.1.1) as follows:  f(u)=F{f(x)} or  f(u)=F{f(x), u}. Given  f(u), the function f(x) can be found by means of the inverse Fourier transform f(x)= 1 √ 2π  ∞ –∞  f(u) e iux du.(11.4.1.2) Formula (11.4.1.2) holds for continuous functions. If f (x)hasa(finite) jump disconti- nuity at a point x =x 0 , then the left-hand side of (11.4.1.2) is equal to 1 2  f(x 0 –0)+f (x 0 +0)  at this point. . integration 11.4. Various Forms of the Fourier Transform 11.4.1. Fourier Transform and the Inverse Fourier Transform 11.4.1-1. Standard form of the Fourier transform. The Fourier transform is defined as. Inversion Formulas for Some Functions 11.2.2-1. Convolution theorem. Main properties of the Laplace transform. 1 ◦ .Theconvolution of two functions f(x)andg(x)isdefined as an integral of the form  x 0 f(t)g(x. instead of passing to the limit. 11.3. Mellin Transform 11.3.1. Mellin Transform and the Inversion Formula 11.3.1-1. Mellin transform. Suppose that a function f(x)isdefined for positive x and satisfies