444 INTEGRAL TRANSFORMS TABLE 11.4 Main properties of the Fourier transform No. Function Fourier transform Operation 1 af 1 (x)+bf 2 (x) a  f 1 (u)+b  f 2 (u) Linearity 2 f(x/a), a > 0 a  f(au) Scaling 3 x n f(x); n = 1, 2, i n  f (n) u (u) Differentiation of the transform 4 f  xx (x) –u 2  f(u) Differentiation 5 f (n) x (x) (iu) n  f (u) Differentiation 6  ∞ –∞ f 1 (ξ)f 2 (x – ξ) dξ  f 1 (u)  f 2 (u) Convolution For brevity, we rewrite formula (11.4.1.2) as follows: f(x)=F –1 {  f(u)} or f (x)=F –1 {  f(u), x}. 11.4.1-2. Asymmetric form of the Fourier transform. Alternative Fourier transform. 1 ◦ . Sometimes it is more convenient to define the Fourier transform by ˇ f(u)=  ∞ –∞ f(x)e –iux dx. In this case, the Fourier inversion formula reads f(x)= 1 2π  ∞ –∞ ˇ f(u)e iux du. 2 ◦ . Sometimes the alternative Fourier transform is used (and called merely the Fourier transform), which corresponds to the renaming e –iux  e iux on the right-hand sides of (11.4.1.1) and (11.4.1.2). 11.4.1-3. Convolution theorem. Main properties of the Fourier transforms. 1 ◦ .Theconvolution of two functions f(x)andg(x)isdefined as f(x) ∗g(x) ≡ 1 √ 2π  ∞ –∞ f(x – t)g(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 F  f(x) ∗g(x)  = F  f(x)  F  g(x)  . 2 ◦ . The main properties of the correspondence between functions and their Fourier trans- forms are gathered in Table 11.4. 11.4. VARIOUS FORMS OF THE FOURIER TRANSFORM 445 11.4.1-4. n-dimensional Fourier transform. The Fourier transform admits n-dimensional generalization:  f(u)= 1 (2π) n/2  R n f(x)e –i(u⋅x) dx,(u ⋅ x)=u 1 x 1 + ···+ u n x n ,(11.4.1.3) where f(x)=f (x 1 , , x n ),  f(u)=f (u 1 , , u n ), and dx = dx 1 dx n . The corresponding inversion formula is f(x)= 1 (2π) n/2  R n  f(u)e i(u⋅x) du, du = du 1 du n . The Fourier transform (11.4.1.3) is frequently used in the theory of linear partial differ- ential equations with constant coefficients (x R n ). 11.4.2. Fourier Cosine and Sine Transforms 11.4.2-1. Fourier cosine transform. 1 ◦ . Let a function f(x) be integrable on the semiaxis 0 ≤ x < ∞.TheFourier cosine transform is defined by  f c (u)=  2 π  ∞ 0 f(x)cos(xu)dx, 0 < u < ∞.(11.4.2.1) For given  f c (u), the function can be found by means of the Fourier cosine inversion formula f(x)=  2 π  ∞ 0  f c (u)cos(xu)du, 0 < x < ∞.(11.4.2.2) The Fourier cosine transform (11.4.2.1) is denoted for brevity by  f c (u)=F c  f(x)  . 2 ◦ . It follows from formula (11.4.2.2) that the Fourier cosine transform has the property F 2 c = 1. Some other properties of the Fourier cosine transform: F c  x 2n f(x)  =(–1) n d 2n du 2n F c  f(x)  , n = 1, 2, ; F c  f  (x)  =–u 2 F c  f(x)  . Here, f (x) is assumed to vanish sufficiently rapidly (exponentially) as x →∞.Forthe second formula, the condition f  (0)=0 is assumed to hold. Parseval’s relation for the Fourier cosine transform:  ∞ 0 F c  f(x)  F c  g(x)  du =  ∞ 0 f(x)g(x) dx. There are tables of the Fourier cosine transform (see Section T3.3 and the references listed at the end of the current chapter). 3 ◦ . Sometimes the asymmetric form of the Fourier cosine transform is applied, which is given by the pair of formulas ˇ f c (u)=  ∞ 0 f(x)cos(xu)dx, f(x)= 2 π  ∞ 0 ˇ f c (u)cos(xu) du. 446 INTEGRAL TRANSFORMS 11.4.2-2. Fourier sine transform. 1 ◦ . Let a function f(x) be integrable on the semiaxis 0 ≤ x < ∞.TheFourier sine transform is defined by  f s (u)=  2 π  ∞ 0 f(x)sin(xu) dx, 0 < u < ∞.(11.4.2.3) For given  f s (u), the function f(x) can be found by means of the inverse Fourier sine transform f(x)=  2 π  ∞ 0  f s (u)sin(xu) du, 0 < x < ∞.(11.4.2.4) The Fourier sine transform (11.4.2.3) is briefly denoted by  f s (u)=F s  f(x)  . 2 ◦ . It follows from formula (11.4.2.4) that the Fourier sine transform has the property F 2 s = 1. Some other properties of the Fourier sine transform: F s  x 2n f(x)  =(–1) n d 2n du 2n F s  f(x)  , n = 1, 2, ; F s  f  (x)  =–u 2 F s  f(x)  . Here, f (x) is assumed to vanish sufficiently rapidly (exponentially) as x →∞.Forthe second formula, the condition f (0)=0 is assumed to hold. Parseval’s relation for the Fourier sine transform:  ∞ 0 F s  f(x)  F s  g(x)  du =  ∞ 0 f(x)g(x) dx. There are tables of the Fourier cosine transform (see Section T3.4 and the references listed at the end of the current chapter). 3 ◦ . Sometimes it is more convenient to apply the asymmetric form of the Fourier sine transform defined by the following two formulas: ˇ f s (u)=  ∞ 0 f(x)sin(xu) dx, f(x)= 2 π  ∞ 0 ˇ f s (u)sin(xu) du. 11.5. Other Integral Transforms 11.5.1. Integral Transforms Whose Kernels Contain Bessel Functions and Modified Bessel Functions 11.5.1-1. Hankel transform. 1 ◦ .TheHankel transform is defined as follows:  f ν (u)=  ∞ 0 xJ ν (ux)f(x) dx, 0 < u < ∞,(11.5.1.1) 11.5. OTHER INTEGRAL TRANSFORMS 447 where ν >– 1 2 and J ν (x) is the Bessel function of the first kind of order ν (see Section SF.6). For given  f ν (u), the function f(x) can be found by means of the Hankel inversion formula f(x)=  ∞ 0 uJ ν (ux)  f ν (u) du, 0 < x < ∞.(11.5.1.2) Note that if f(x)=O(x α )asx → 0,whereα + ν + 2 > 0,andf (x)=O(x β )asx →∞, where β + 3 2 < 0, then the integral (11.5.1.1) is convergent. The inversion formula (11.5.1.2) holds for continuous functions. If f (x)hasa(finite) jump discontinuity at a point x = x 0 , then the left-hand side of (11.5.1.2) is equal to 1 2 [f(x 0 – 0)+f(x 0 + 0)] at this point. For brevity, we denote the Hankel transform (11.5.1.1) by  f ν (u)=H ν  f(x)  . 2 ◦ . It follows from formula (11.5.1.2) that the Hankel transform has the property H 2 ν = 1. Other properties of the Hankel transform: H ν  1 x f(x)  = u 2ν H ν–1  f(x)  + u 2ν H ν+1  f(x)  , H ν  f  (x)  = (ν – 1)u 2ν H ν+1  f(x)  – (ν + 1)u 2ν H ν–1  f(x)  , H ν  f  (x)+ 1 x f  (x)– ν 2 x 2 f(x)  =–u 2 H ν  f(x)  . The conditions lim x→0  x ν f(x)  = 0, lim x→0  x ν+1 f  (x)  = 0, lim x→∞  x 1/2 f(x)  = 0, lim x→∞  x 1/2 f  (x)  = 0 are assumed to hold for the last formula. Parseval’s relation for the Hankel transform:  ∞ 0 uH ν  f(x)  H ν  g(x)  du =  ∞ 0 xf(x)g(x) dx, ν >– 1 2 . 11.5.1-2. Meijer transform. The Meijer transform is defined as follows: ˆ f μ (s)=  2 π  ∞ 0 √ sx K μ (sx)f(x) dx, 0 < s < ∞, where K μ (x) is the modified Bessel function of the second kind (the Macdonald function) of order μ (see Section SF.7). For given  f μ (s), the function f (x) can be found by means of the Meijer inversion formula f(x)= 1 i √ 2π  c+i∞ c–i∞ √ sx I μ (sx) ˆ f μ (s) ds, 0 < x < ∞, where I μ (x) is the modified Bessel function of the first kind of order μ (see Section SF.7). For the Meijer transform, a convolution is defined and an operational calculus is developed. 448 INTEGRAL TRANSFORMS 11.5.1-3. Kontorovich–Lebedev transform. The Kontorovich–Lebedev transform is introduced as follows: F (τ)=  ∞ 0 K iτ (x)f(x) dx, 0 < τ < ∞, where K μ (x) is the modified Bessel function of the second kind (the Macdonald function) of order μ (see Section SF.7) and i = √ –1. For given F (τ), the function can be found by means of the Kontorovich–Lebedev inversion formula f(x)= 2 π 2 x  ∞ 0 τ sinh(πτ)K iτ (x)F (τ) dτ , 0 < x < ∞. 11.5.1-4. Y -transform. The Y -transform is defined by F ν (u)=  ∞ 0 √ ux Y ν (ux)f(x) dx, where Y ν (x) is the Bessel function of the second kind of order ν. Given a transform F ν (u), the inverse Y -transform f(x) is found by the inversion formula f(x)=  ∞ 0 √ ux H ν (ux)F ν (u) du, where H ν (x) is the Struve function, which is defined as H ν (x)= ∞  j=0 (–1) j (x/2) ν+2j+1 Γ  j + 3 2  Γ  ν + j + 3 2  . 11.5.2. Summary Table of Integral Transforms. Areas of Application of Integral Transforms 11.5.2-1. Summary table of integral transforms. Table 11.5 summarizes the integral transforms considered above and also lists some other integral transforms; for the constraints imposed on the functions and parameters occurring in the integrand, see the references given at the end of this section. 11.5.2-2. Areas of application of integral transforms. Integral transforms are widely used for the evaluation of integrals, summation of series, and solution of various mathematical equations and problems. In particular, the application of an appropriate integral transform to linear ordinary differential, integral, and difference equations reduces the problem to a linear algebraic equation for the transform; and linear partial differential equations are reduced to an ordinary differential equation. 11.5. OTHER INTEGRAL TRANSFORMS 449 TABLE 11.5 Summary table of integral transforms Integral transform Definition Inversion formula Laplace transform  f(p)=  ∞ 0 e –px f(x) dx f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) dp Laplace– Carlson transform  f(p)=p  ∞ 0 e –px f(x) dx f(x)= 1 2πi  c+i∞ c–i∞ e px  f(p) p dp Two-sided Laplace transform  f ∗ (p)=  ∞ –∞ e –px f(x) dx f(x)= 1 2πi  c+i∞ c–i∞ e px  f ∗ (p) dp Fourier transform  f(u)= 1 √ 2π  ∞ –∞ e –iux f(x) dx f(x)= 1 √ 2π  ∞ –∞ e iux  f(u) du Fourier sine transform  f s (u)=  2 π  ∞ 0 sin(xu)f(x) dx f(x)=  2 π  ∞ 0 sin(xu)  f s (u) du Fourier cosine transform  f c (u)=  2 π  ∞ 0 cos(xu)f(x) dx f(x)=  2 π  ∞ 0 cos(xu)  f c (u) du Hartley transform  f h (u)= 1 √ 2π  ∞ –∞ (cos xu +sinxu)f(x) dx f(x)= 1 √ 2π  ∞ –∞ (cos xu +sinxu)  f h (u) du Mellin transform  f(s)=  ∞ 0 x s–1 f(x) dx f(x)= 1 2πi  c+i∞ c–i∞ x –s  f(s) ds Hankel transform  f ν (w)=  ∞ 0 xJ ν (xw)f (x) dx f(x)=  ∞ 0 wJ ν (xw)  f ν (w) dw Y -transform F ν (u)=  ∞ 0 √ ux Y ν (ux)f(x) dx f(x)=  ∞ 0 √ ux H ν (ux)F ν (u) du Meijer transform (K-transform)  f (s)=  2 π  ∞ 0 √ sx K ν (sx)f(x) dx f(x)= 1 i √ 2π  c+i∞ c–i∞ √ sx I ν (sx)  f(s) ds Bochner transform  f(r)=  ∞ 0 J n/2–1 (2πxr)G(x,r)f(x) dx, G(x, r)=2πr(x/r) n/2 , n= 1, 2, f(x)=  ∞ 0 J n/2–1 (2πrx)G(r, x)  f(r) dr Weber transform F a (u)=  ∞ a W ν (xu, au)xf(x) dx, W ν (β, μ)≡ J ν (β)Y ν (μ)–J ν (μ)Y ν (β) f(x)=  ∞ 0 W ν (xu, au) J 2 ν (au)+Y 2 ν (au) uF a (u) du Hardy transform F (u)=  ∞ 0 C ν (xu)xf(x) dx, C ν (z)≡ cos(πp)J ν (z)+sin(πp)Y ν (z) f(x)=  ∞ 0 Φ(xu)uF (u) du Φ(z)= ∞  n=0 (–1) n (z/2) ν+2p+2n Γ(p+n+1)Γ(ν+p+n+1) Kontorovich– Lebedev transform F (τ )=  ∞ 0 K iτ (x)f(x) dx f(x)= 2 π 2 x  ∞ 0 τ sinh(πτ)K iτ (x)F (τ) dτ Meler–Fock transform  F (τ )=  ∞ 1 P – 1 2 +iτ (x)f(x) dx f(x)=  ∞ 0 τ tanh(πτ)P – 1 2 +iτ (x)  F (τ ) dτ Euler transform of the 1st kind* F (x)= 1 Γ(μ)  x a f(t) dt (x – t) 1–μ 0 <μ <1, x > a f(x)= 1 Γ(1 – μ) d dx  x a F (t) dt (x – t) μ 450 INTEGRAL TRANSFORMS TABLE 11.5 (continued) Summary table of integral transforms Integral transform Definition Inversion formula Euler transform of the 2nd kind* F (x)= 1 Γ(μ)  a x f(t) dt (t – x) 1–μ 0 <μ <1, x < a f(x)=– 1 Γ(1 – μ) d dx  a x F (t) dt (t – x) μ Gauss transform** F (x)= 1 √ πa  ∞ –∞ exp  – (x – t) 2 a  f(t) dt f(x)=exp  – a 4 d 2 dx 2  F (x) Hilbert transform***  F (s)= 1 π  ∞ –∞ f(x) x – s dx f(x)=– 1 π  ∞ –∞  F (s) s – x ds NOTATION: i= √ –1, J μ (x)andY μ (x) are theBessel functions of thefirst and the second kind, respectively; I μ (x)andK μ (x) are the modified Bessel functions of the first and the second kind, respectively; P μ (x)isthe Legendre spherical function of the second kind; and H μ (x) is the Struve function (see Subsection 11.5.1-4). R EMARKS. * The Euler transform of the first kind is also known as Riemann–Liouville integral (the left fractional integral of order μ or, for short, the fractional integral). The Euler transform of the second kind is also called the right fractional integral of order μ. ** If a =4, the Gauss transform is called the Weierstrass transform. In the inversion formula, the exponential is represented by an operator series: exp  k d 2 dx 2  ≡ 1 + ∞  n=1 k n n! d 2n dx 2n . *** In the direct and inverse Hilbert transforms, the integrals are understood in the sense of the Cauchy principal value. Table 11.6 presents various areas of application of integral transforms with literature references. Example. Consider the Cauchy problem for the integro-differential equation dy dx +  x 0 K(x – t)y(t) dt = f(x)(0 ≤ t < ∞)(11.5.2.1) with the initial condition y = a at t = 0.(11.5.2.2) Multiply equation (11.5.2.1) by e –px and then integrate with respect to x from zero to infinity. Using properties 7 and 12 of the Laplace transform (Table 11.1) and taking into account the initial condition (11.5.2.2), we obtain a linear algebraic equation for the transform y(p): py(p)–a +  K(p)y(p)=  f(p). It follows that y(p)=  f(p)+a p +  K(p) . By the inversion formula (11.2.1.2), the solution to the original problem (11.5.2.1)–(11.5.2.2) is found in the form y(x)= 1 2πi  c+i∞ c–i∞  f(p)+a p +  K(p) e px dp, i 2 =–1.(11.5.2.3) Consider the special case of a = 0 and K(x)=cos(bx). From row 10 of Table 11.2 it follows that  K(p)= p p 2 + b 2 . Rearrange the integrand of (11.5.2.3):  f(p) p +  K(p) = p 2 + b 2 p(p 2 + b 2 + 1)  f(p)=  1 p – 1 p(p 2 + b 2 + 1)   f(p). In order to invert this expression, let us use the convolution theorem (see formula 16 of Subsection T3.2.1) as well as formulas 1 and 28 for the inversion of rational functions, Subsection T3.2.1. As a result, we arrive at the solution in the form y(x)=  x 0 b 2 +cos  t √ b 2 + 1  b 2 + 1 f(x – t) dt. . transforms. Integral transforms are widely used for the evaluation of integrals, summation of series, and solution of various mathematical equations and problems. In particular, the application of. Table of Integral Transforms. Areas of Application of Integral Transforms 11.5.2-1. Summary table of integral transforms. Table 11.5 summarizes the integral transforms considered above and also. other integral transforms; for the constraints imposed on the functions and parameters occurring in the integrand, see the references given at the end of this section. 11.5.2-2. Areas of application of integral