PROBLEMS 473 where the degeneracy, gn, and the eigenfrequencies, wn, are given as Note: Interested students can obtain the eigenfrequencies and the degen- eracy by solving the wave equation for the massless conformal scalar field: 1 (n-2) 4(n-1) O@(f,t) + -~ R@(7,t) = 0, where n is the dimension of spacetime, R is the scalar curvature, and 0 is the d' Alembert (wave) operator !J = 9pvap3v, (15.235) where a, stands for the covariant derivative. Use the separation of variables method and impose the boundary condition Q, = finite on the sphere. For this problem spacetime dimension n is 3 and for a sphere of constant radius, &, the curvature scalar is 2/R& 15.4 Using asymptotic series evaluate the logarithmic integral dt O<z<l. Hint: Use the substitutions t = e-" and a = -lnx, a > 0, and integrate by parts successively to write the series 1 I! 21 (n - I)! a a2 a3 an - +- +(-I)"-'- where so that 15.5 massless conformal scalar field with thermal spectrum can be written as In a closed Einstein universe the renormalized energy density of a 474 INFINITE SERIES where & is the constant radius of the universe, T is the temperature of the radiation, and (2x21i$) is the volume of the universe. The second term (-) inside the square brackets is the well-known renormalized quantum vacuum energy, that is, the Casimir energy for the Einstein universe. First find the high and low temperature limits of (p}ren. and then obtain the flat spacetime limit & -+ 03. 15.6 places: tic 240& Without using a calculator evaluate the following sum to five decimal 00. n=6 How many terms did you have to add? 15.7 Check the convergence of the series 15.8 Find the interval of convergence for the series 15.9 Evaluate the sums cr=o an cos ne (b) C,"==, an sin d, a is a constant Hint: Try using complex variables. 15.10 Verify the following Taylor series: X" 00 ex = C 12? for all z n=O and PROBLEMS 475 15.11 Find the first three nonzero terms of the following Taylor series: a) f(x) = x3 + 2x + 2 about z = 2 b) f(x) = e2= cos x about x = 0 15.12 Another important consequence of the Lorentz transformation is the formula for the addition of velocities, where the velocities measured in the K and R frames are related by the formula dx &1 dt di where u1 = - and El = - are the velocities measured in the K and I? frames, respectively, and R is moving with respect to K with velocity u along the common direction of the x- and Z-axes. Using the binomial formula find an appropriate expansion of the above formula and show that in the limit of small velocities this formula reduces to the well-known Galilean result 15.13 as In Chapter 10 we have obtained the formulas for the Doppler shift w = rw’(1 - PCOS8) tan@ = sinQ/y(cose -p), where 8, 8’ are the angles of the wave vectors k and k with respect to the relative velocity 37’ of the source and the observer. Find the nonrelativistic limit of these equations and interpret your results. 15.14 Given a power series i- show that the differentiated and integrated series will have the same radius of convergence. 15.15 Expand 1 h(x) = tanhz - - 2 476 INFINITE SERIES as a power series of x. 15.16 Find the sum 1 X x2 x4 1-2 2-3 3.4 4.5 g(x) = - + - + - + - + Hint: First try to convert into geometric series. h(1-x) . 1 1-z g(x) = - + - 1 Answer: [ x 22 15.17 Using the geometric series evaluate the sum W C n3xn n= 1 exactly for the interval 1x1 < 1, then expand your answer in powers of X. 15.18 By using the Euler-Maclaurin sum formula evaluate the sum W C n3xn n=l Show that it agrees with the expansion found in Problem 15.17. 16 INTEGRAL TRANSFORMS Integral transforms are among the most versatile mathematical tools. Their applications range from solution of differential equations to evaluation of def- inite integrals and from solution of systems of coupled differentia1 equations to integral equations. They can even he used for defining differintegrals, that is, fractional derivatives and integrals (Chapter 14). In this chapter, after a general introduction we mainly discuss two of the most frequently used inte- gral transforms, the Fourier and the Laplace transforms, their properties, and techniques. Commonly encountered integral transforms allow us to relate two functions through the integral (16.1) where g(a) is called the integral transform of f(t) with respect to the kernel ~(a, t). These transformations are also linear, that is, if the transforms exist, then one can write (16.3) 4 77 478 INTEGRAL TRANSFORMS and b C9l (a) = 1 Id1 (t>l4a, t)dt, (16.4) where c is a constant. Integral transforms can also be shown as an operator: where the operator =€(a,t) is defined as (16.6) We can now show the inverse transform as 16.1 SOME COMMONLY ENCOUNTERED INTEGRAL TRANSFORMS Fourier transforms are among the most commonly encountered integral trans- forms. They are defined as loo (16.8) g(a) = - 1 f(t)ei"tdt. Because the kernel of the Fourier transform is also used in defining waves, they are generally used in the study of wave phenomena. Scattering of X-rays from atoms is a typical example. The Fourier transform of the amplitude of the scattered waves gives the electron distribution. Fourier cosine and sine transforms are defined as d% -cc (16.10) Other frequently used kernels are e-at, tJn(at), and ta-'. (16.11) The Laplace transform is defined as ( 16.12) DERIVATION OF THE FOURIER INTEGRAL 479 and it is very useful in finding solutions of systems of ordinary differential equations by converting them into a system of algebraic equations. The Han- kel or Fourier-Bessel transform is defined as and it is usually encountered in potential energy calculations in cylindrical coordinates. Another useful integral transform is the Mellin transform: (16.14) The Mellin transform is useful in the reconstruction of functions" from power series expansions. Weierstrass function is defined as "Weierstrass-type 00 (16.15) n=O where a and b are constants. It has been proven that, provided 0 < b < 1, a > 1, and ab > 1, the Weierstrass function has the interesting property of being continuous but nowhere differentiable. These interesting functions have found widespread use in the study of earthquakes, rupture, financial crashes, etc. (Gluzman and Sornette). 16.2 DERIVATION OF THE FOURIER INTEGRAL 16.2.1 Fourier Series Fourier series are very useful in representing a function in a finite interval, like [0,27r] or [-L, L], or a periodic function in the infinite interval (-CO,CO). We now consider a nonperiodic function in the infinite interval (-o,co). Physically this corresponds to expressing an arbitrary signal in terms of sine and cosine waves. We first consider the trigonometric Fourier expansion of a sufficiently smooth function in the finite interval [-L, L] as Fourier expansion coefficients a, and b, are given as (16.17) (16.18) 480 [NTEGRAL TRANSFORMS Substituting an and b, explicitly into the Fourier series and using the trigone metric identity cos(a - b) = cos a cos b + sin a sin b, (16.19) we get Since the eigenfrequencies are given as w = y, where n = 0,1,2,. . . , the distance between two neighboring eigenfrequencies is (16.21) R aw= L Using Equation (16.21) we can write f(z) = IL f(t)dt + 12 A wIW f(t)ccsw(t -x)dt. (16.22) -00 n=~ 2L -L We now take the continuum limit, L + 03, where we can make the replace ment 00 A w t lmdw. n=l (16.23) Thus the Fourier integral is obtained as f(.) = a 1°0 dw 1: f(t) cosw(t - z)dt. (16.24) In this expression we have assumed the existence of the integral 00 1, f (16.25) For the Fourier integral of a function to exist, it is sufficient for the integral I-", If(t)l dt to be convergent. We can also write the Fourier integral in exponential form. Using the fact that sin w(t - x) is an odd function with respect to w, we can write 03 (16.26) Also since cosw(t - x) is an even function with respect to w, we can extend the range of the w integral in the Fourier integral to (-00, 00) as 1" l, dw 1, f(t) sinw(t - x)dt = 0. 00 f(.) = & ./" dw .I_, f(t) cosw(t - z)dt. (16.27) -~ 00 FOURIER AND INVERSE FOURIER TRANSFORMS 481 We now multiply Equation (16.26) by z and then add Equation (16.27) to obtain the exponential form of the Fourier integral as f(.) = 2 1 dwe-iWx 1, f weiwt& (16.28) where w is a parameter; however, in applications to waves it is the angular frequency. W 03 27r oo 16.2.2 Dirac-Delta Function Let us now write the Fourier integral as where we have interchanged the order of integration. The expression inside the curly brackets is nothing but the Dirac-delta function: (16.30) which has the following properties: 6(. - u) = 0, ( 5 # a), (16.31) 6(z - a)& = 1, (16.32) - a)f(.)d. = f(a), (16.33) 00 1, where f(z) is continuous at 2 = a. 16.3 FOURIER AND INVERSE FOURIER TRANSFORMS We write the Fourier integral theorem [Eq. (16.28)] as We now define the Fourier transform of f(t) as 1 Po0 where the inverse Fourier transform is defined as (16.34) (16.35) 1" f(t) = - / g(w)eciwtdw. 6 co 482 INTEGRAL TRANSFORMS Fig. 16.1 Wave train with N = 5 16.3.1 Fourier Sine and Cosine Transforms When f(t) is an even function we can write fc( ) = fc(z). Using the identity (16.36) eiWt - - cos wt + i sin wt, (16.37) we can also write gc(w> = - fc(t) (cos wt + i sin wt) dt. (16.38) Considering that sin wt is an odd function with respect to t, the Fourier cosine transform is obtained as gc(w) = &yfc(I)cosLltdt. (16.39) & 1, The inverse Fourier cosine transform is given as fc(t) = Ely gc(w) coswtdw. Similarly, for an odd function we can write f3( ) = -fs(.). From the Fourier integral we obtain its Fourier sine transform as gs(w) = Ely f,(t)sinwtdt, and its inverse Fourier sine transform is fs(z) = EJr(mgs(w)sinwzdw. (16.40) (16.41) (16.42) (16.43) [...]... given in terms of the Bromwich integral as (16.104) where y is real and s is now a complex variable The contour for the above integral is an infinite straight line passing through the point y and parallel t o the imaginary axis in the complex s-plane y is chosen such that all the singularities of e s t f ( s ) are t o the left of the straight line For t > 0 we can close the contour with a n infinite... left-hand side of the line The above integral can now be evaluated by using the residue theorem to find the inverse Laplace transform The Bromwich integral is a powerful tool for inverting complicated Laplace transforms when other means prove inadequate However, in practice using the fact that Laplace transforms are linear and with the help of some basic theorems we can generate many of the inverses... e-lCt) 2 INVERSE LAPLACE TRANSFORMS 493 F g 16.4 Heavyside step function i and 1 2 F ( t ) = sinh kt = - (elct - e-lct) X can be found by using the fact that is a linear operator as S L {cosh k t } = k L {sinh k t } = s2 - k2 ~ where s (16.109) (16.110) ’ > k for both 5 Using the relations cos kt = cosh ikt sin k t = -i sinh k t , and we can find the Laplace transforms of the cosine and the sine functions...483 FOURIER A N D INVERSE FOURIER TRANSFORMS Example 16.1 Fourier analysis of finite wave train: We now find the Fourier transform of a finite wave train, which is given as (16.44) For N = 5 this wave train is shown in Figure 16.1 I Since f ( t ) is an odd function we find its Fourier sine transform as &[ 2 sin (wO - w)EL gs(w) = wo 2 (WO - W ) - - + sin (wo w) 2(wo+w) (16.45) For frequencies... k ) and G ( k )are the Fourier transforms of f(x) and g(z), respectively Proof: To prove these theorems we make the k + -k change in the Fourier transform of g(x): Srn G(-k) = - 6 g(z)e-i"h (16.73) Poi) Multiplying the integral in Equation (16.73) with F ( k ) and integrating it over k in the interval (-m, co) we get la 00 d k F ( k ) G ( - k )= / 00 1 " dkF(k)- 6 -00 hg(z)e-i" (16.74) -00 Assuming... sintx Thus we find O0 sintx dx (16.195) (16.196) 7r L { F ( t ) } - = 2s (16.197) LAPLACE TRANSFORM OF A DERIVATIVE 503 Finding the inverse Laplace transform of this gives us the value of the definite integral as ll F ( t ) = -, t > 0 2 (16.198) 16.7 LAPLACE TRANSFORM OF A DERIVATIVE One of the main applications of Laplace transforms is to differential equations In particular, systems of ordinary linear... Completing the square in the denominator we write m m (16.219) 506 INTEGRAL TRANSFORMS For weak damping, b2 we find X (s) = xo = xo < 4 k m , the last term S+-& (s+ b (16.220) +w; & J 2 b s+-+- 2m (s+&) = xo S+- is positive Calling this w:, 2 b 2m +w,2 - 2m +W; Taking the inverse Laplace transform of X ( s )we find the final solution as (16.221) Check that this solution satisfies the given initial... ) = 0 + After integration, we find f (s) as ( 16.238) (16.239) To find the inverse we write the binomial expansion of f (s) as C f(s)= [ (16.240) 1 2s2 = - 1 + S + (- +-.I 1.3 222!s4 1)”(2n)! (2nn!)2 s2n (16.241) Using t h e fact that Laplace transforms are linear, we find the inverse as (16.242) Using the condition y(0) = 1, we determine the constant a c as one s This solution is nothing but the zeroth-order... (16.191) ~ " I inverse Laplace transform of which can be found easily as L-' {f(s)}= -a e-2t + cos f i k t - -sin h k t k , a = -k2/((2k2 + 4) (16.192) Example 16.6 Definite integrals and Laplace transforms: We can also use integral transforms to evaluate some definite integrals Let us consider (16.193) Taking the Laplace transform of both sides we get = [la 1 dte-st sin(tx) dx The quantity inside the... the initial shape of the wave as E(0,t ) = W ) , (16.229) L ( E ( 0 , t ) )= C ] = f ( s ) (16.230) we determine c1 as Thus, with the given initial conditions, the Laplace transform of the solution is given as L {E(x,t)} e= xf(s) (16.231) Using theorem I1 we can find the inverse Laplace transform, and the final solution is obtained as This is a wave moving along the positive z-axis with velocity u and . defined as and it is usually encountered in potential energy calculations in cylindrical coordinates. Another useful integral transform is the Mellin transform: (16 .14) The Mellin transform. function in the infinite interval (-CO,CO). We now consider a nonperiodic function in the infinite interval (-o,co). Physically this corresponds to expressing an arbitrary signal in terms. for the above integral is an infinite straight line passing through the point y and parallel to the imaginary axis in the complex s-plane. y is chosen such that all the singularities of