Twodimensional Fourier transformations and Mordell integrals


 Λήδα Καρράς
 3 μήνες πριν
 Προβολές:
Transcript
1 Twodimensional Fourier transformations and Mordell integrals Martin Nicholson Several Fourier transformations of functions of one and two variables are evaluated and then used to derive some integral and series identities. It is shown that certain twodimensional Mordell integrals factorize into roduct of two integrals and that the square of the absolute value of the Mordell integral can be reduced to a single onedimensional integral. Some connections to ellitic functions and lattice sums are discussed. I. Introduction: selfrecirocal Fourier transformations Define the cosine and sine Fourier transformations by the usual formulas f c (t = f s (t = f(x cos tx dx, ( f(x sin tx dx. ( Functions that are equal to their own cosine Fourier transform, i.e. that satisfy the equation f(x = f c (x, are called selfrecirocal functions of the first kind, and functions that are equal to their own sine Fourier transform f(x = f s (x, are called selfrecirocal functions of the second kind[]. Some examles of the functions of the first kind include x, x x, + And here are some functions of the second kind x sinh x, sinh 6 x x, x, x, 4 x sinh x x, x x cos. ( sinh x x cos. (4 The first three functions of ( and the first two functions of (4 were known to Ramanujan and their detailed study can be found in the book []. The third function in (4 is taken from the article [] where many other hyerbolic self recirocal functions are given along with a general method for generating them. The last two functions in ( and the last function in (4 aear to be new. One can show that ( are the αx x+c. only self recirocal functions of the form There is a well known general recie to find self recirocal functions ([4, ch. 9]. Since (f c c = f, the sum f(x + f c (x is a selfrecirocal function of the first kind for an arbitrary function f(x. Obviously this aroach works also for functions of the second kind. It might seem that this settles the question of finding all selfrecirocal functions comletely. However this is not so because this aroach is not helful in finding interesting articular selfrecirocal functions.
2 It is much more gratifying to now that the functions in ( are selfrecirocal as oosed to knowing that the function e x + + x is selfrecirocal. A more useful general theory suitable for these uroses of finding articular transformations has been develoed by Goodseed, Hardy and Titchmarsh (see [4] for a nice account of this theory. One might ask, what are these articular transformations useful for? The answer is they lead to some interesting integral and series transformation formulas, among other things. For examle, Hardy and Ramanujan [,5] used self recirocal functions to obtain transformation formulas such as e α x αx dx = β αx α dx = β αx e x α sinh αx sinh αx xe x dx = β e y dy, αβ =, (5 βy βy βy e y dy, αβ =, (6 sinh βy sinh βy ye y dy, αβ =. (7 Another tye of identities are obtained by alication of the Poisson summation formula, which for an even function φ(x can be stated in the symmetric form [4] α φ(αn = β φ c (βn, αβ =. (8 n= Similarly, for an odd function ψ(x α χ(nψ(αn = β χ(nψ s (βn, αβ =, (9 n= where χ(n = sin n is a rimitive character of modulus 4. For examle, alication of (8 to the first function in ( gives α αn = β, αβ =. ( βn Let q = e α be the base of ellitic functions with modulus k, k = k the comlementary modulus and K = K(k, K = K(k the comlete ellitic integrals of the first kind. Then [6, ch..6] q = e β is the base of ellitic functions with modulus k and K = αn. ( K So ( is nothing but q = e K in the more familiar notation of the theory of ellitic functions. Functions (,(4 imly certain symmetric relations for the Lerch zeta function ([], ch For examle the fourth function in (4 leads to the identity n sin( n + = sin( n q n + q, q =, < <. (
3 II. Functions of two variables One may also consider self recirocal Fourier functions of two variables. Aart from the noninteresting factorizable functions of this form there are quite nontrivial functions. To find some of them we use the following observation: If f(x, y = f(y, x and f(x, y cos axdx = g(a, y = g(y, a, (in other words, if artial Fourier transform of a symmetric function is symmetric then f(x, y is a selfrecirocal Fourier function of two variables, i.e. Examle : Since ([7], formula.98.8 f(x, y cos ax cos bydxdy = f(a, b. sin xy sinh( x cos axdx = sinh( y ( y + ( a we get a air of selfrecirocal Fourier transformations cos ax cos by x + y dxdy = a + a, ( sin xy sinh( x sinh( y cos ax cos bydxdy = sin Though not a self recirocal function, note the curious transformation x ab sinh( a sinh( b. (4 y cos ax cos by dxdy = sin ab sinh a sinh (5 b. More selfrecirocal functions of one and two variables can be found in []. Poisson summation formula (8 is easily generalized to even functions of two variables as follows where αβ m, φ(αm, βn = γδ φ c (t, s = m, φ c (γm, δn, αγ = βδ =, (6 φ(x, y cos tx cos sy dxdy. It is instructive to see what haens if (6 is alied to (5. Straightforward calculation shows that αβ m, cos αβmn αm βn = γδ m, sin γδmn sinh γm sinh αγ = βδ =. δn,
4 Here it is assumed that the terms with m = or n = on the RHS of are understood as the limits lim m, lim n. Setting δ = α, γ = β and making the relacement α α, β β one obtains (care should be taken to simlify the sum on the right nα It is known that [6, ch..75] nβ = + 4 n= n= αn sinh nα + 4 n K(K E = sinh nα, n= 4 βn, αβ =. (7 sinh nβ with the same notations as in ( and E = E(k comlete ellitic integral of the second kind. Therefore (7 is Legendre s relation EK + E K KK = in disguise. Hyerbolic functions rovide many other transformations. Let s start with the calculation of the integral By formula.98. from [7]: so finally J = = = = J = cos by (y/ dy cos by (y/ a a x (y/ x cos ax cos by dxdy. cos ax dx a y y + a ( y (y/ dy a y + y a b a b cos ab a (b, x (y/ cos ax cos by dxdy = a b a cos bydy cos ab b a (8 b. We see that the right hand side is the original function (taken with the minus sign u to an additional term, which a factorizable function. Alying Poisson summation (6 to (8 one finds m= αm βn = m= αm αm βn, αβ =. (9 βn (9 is equivalent to the modulus transformation of Landen s transform, i.e. ( + k ( + k = in the notation of the book [6]. Indeed, if α = K(k K(k, β = Λ(k Λ(k, then Λ = βn,
5 ( ik dn, k = K ( iλ dn, k = Λ αn αn, βn βn. Since dn ( ik, k = + k, eq. (9 reduces to ( + k( + k =, as required. There is an integral analogous to (8 involving odd functions: More comlicated air of integrals: 4 4 sin xy x (y/ sin ax sin by dxdy = cos ax cos by ( + x( + y = sin xy sin ax sin by ( + x( + y = Eqs. (, ( lead to identities α 4 n= n= n αn sinh ( αn + α α(n+ + n= a b sinh sinh a b sin ab a sinh b sinh a ( cos ab b a sinh b sinh a 5 sin ab b a ( b. + cos ab ( + a ( ( + b, sin ab ( + a ( ( + b. n n ( α sinh ( n /α = + α 4 αn +, ( = χ(n β(n+ n= + n= αn sinh α χ(n n= βn sinh β. (4 Of course it is ossible to derive both ( and (4 from the theory of ellitic functions. However it is not at all obvious that such symmetric relations exist in the first lace and moreover the methodology develoed in this section is useful in derivation of identities that robably can not be obtained from the theory of ellitic functions in a straightforward manner. One such identity is where f β (θ cos θf β(θf β (θ + cos θ sin θ f β(θ = f β (θ = ( βn cos θ + 4 n coth βn βn cos θ βn cos θ. n= III. Twodimensional Mordell integrals Let s multily (8 by e a b and integrate with resect to a and b x (y/ e x y dxdy = a b a be a b dadb cos ab a a b dadb. be
6 6 This can be written in the following symmetrical form αx βy e x y dxdy = αx αx e x dx βy βy e y dy, αβ =. (5 Note the similarity of (5 with the Landen transform (9. Since Mordell integrals can be understood as continous analogs of theta functions [], (5 can be understood as Landen s transform for Mordell integrals. However the factorization on the left side of (5 does not occur because of the function in the integrand (in the discrete case it was ossible to choose the arameters so that didn t have any mixing effect on the two series, so the double series factorized; unfortunately this is not ossible for an integral. Combining (5 with (6 leads to Corollary. where I = x x e x y αx (y/α dxdy = α ( cos nx y n dxdy = x y ( sin nx y n dxdy = x y cos nx dx, I = In analogous manner ( and (7 give Corollary. where I = x sinh x x x x xye x y sin xy αx (y/α dxdy = αx dx αx e x n n I I + ni I, n n I I + ni I, nx sin dx, n >. α sinh αx dx αx xe x ( cos nx y n sin xy n ( xy dxdy = I x y 4 I + I I 4, ( sin nx y n n ( xy dxdy = I x y I4 + I 4, cos nx dx, I 4 = x sinh x x nx sin dx, n >.. (6. (7 Ramanujan showed that integrals I I 4 have closed form exressions for rational n []. So the corresonding twodimensional integrals also have closed form exressions.
7 Examles. ( cos x y dxdy = x y ( sin x y x y 6, dxdy = 4, cos ( x y sin xy xy dxdy = x y 8. It is ossible to calculate even more general integrals. In analogy with Ramanujan s integral analogs of theta functions [] define Φ α,β (θ, φ = cos θx cos φy x y { ex ( αx + βy } dxdy. 7 Then { ( θ αβ ex = α + φ β x θx α x } ( iθ Φ α,β (θ, φ + Φ, α β ex } { x dx α α, iφ β y y φy β ex } { y dy. (8 β Equation (8 generalizes (6. Now one can aly the method develoed by Ramanujan [] to the function Φ α,β (θ, φ. From the definition of Φ α,β (θ, φ it follows that } Φ α,β (θ + i, φ + Φ α,β (θ i, φ = ex { θ α α cos φy θy α y { ex ( β + } y dy. (9 α Now combine (8 and (9 to get { β φ ex β { θ ex α } ( ex } { (θ + α α cos θy φy α y { } α ex 8 + θ θ α } Φ α,β (θ + α, φ + ex { ex ( α + } y dy+ β y y φy β ex { (θ α } { y dy. β α } Φ α,β (θ α, φ = These formulas reduce the roblem to the calculation of onedimensional Mordell integrals. Similar formulas also exist for sin xy sin θx sin φy { Ψ α,β (θ, φ = ex ( αx + βy } dxdy. x y
8 8 IV. Absolute value of the Mordell integral For real α one has for the square of the absolute value of the Mordell integral We can write this as 4 I(α = = = = = 4 e iαx x dx dy e iα(x+y (x + y dy e iαy iαxy x (x + y dxdy e iαxy (x y/ (x + y/ dxdy e iαxy x + y dx sin αy sinh y sinh αy dy. sin αx ( sinh x sinh αx dx = cos αx ( x dx sin αx + x dx. ( Analogous considerations lead to other formulas of similar kind sin αx sinh x sinh αx dx = cos αx x αx dx = coth x αx coth cos αx 4 ( + x( + αx sin αx 4 and to the following curious closed form dx = x x e iαx dx e iαx 4 + x dx, (, ( tanh x tanh αx cos αx dx =. ( Here we give an exlanation for the first equality in ( and for (. For the first, starting from (5 we ut b = αa x and integrate with resect to a from to to obtain For the second, starting from (4 and its sine analog y cos ax cos αay dxdy = sin αa sinh a sinh αa cos αx x αx dx = sin αx sinh x sinh αx dx. sinh( x sinh( y sin ax sin bydxdy = tanh a b tanh cos ab sinh( a sinh( b, (4
9 [by the way (4 imlies the self recirocal function sinh( x sinh( ] we ut b = αa in both, take the sum y of (4 multilied by e iα a and (4 multilied by ie iα a, integrate from to using formulas i cos ax cos αay e iα a da = α (x +α y, to obtain = i sin ax sin αay e iα a da = i a αa tanh tanh α e i i α e i α (x +α y sin xy, i( cos αa + sin αa e iα a da + sinh( a sinh( αa From this it is straightforward to deduce ( and as a byroduct sin αx sinh x sinh αx dx = tanh x tanh αx sin αx dx. Comare ( to the integral of Ramanujan ([8], generalizations are given in [9,] e iα a da. αx x cos αx dx = cos α 4. (5 They both contain trigonometric function of the argument αx and hyerbolic functions of the arguments x and αx. However the crucial difference between them is that the integrand in ( has oles not only at the zeroes of x, but also at the zeroes of αx. Integrals of this sort are related to integrals for the roduct of two hyerbolic selfrecirocal functions studied by Ramanujan ([5], formula (. Put in (8 and ( b = αa and integrate with resect to a. The result is cos αx x αx dx = x αx dx, (6 x αx sin αx x αx dx = sinh x αx sinh dx. (7 x αx 9 Multilying (8 and ( by V. Connection to lattice sums ab and integrating with resect to a and b leads to cos x y dxdy x y = sin x y dxdy x y = x x dx sinh x x dx, (8. (9 The RHS of (8 and (9 contain integral reresentation of certain Dirichlet Lseries, while the LHS are Dlattice sums of Bessel and Neumann functions, as shown below on a formal level. Evaluation of double sums of Bessel functions in terms of Dirichlet Lseries is well known [].
10 Consider the double integral on the LHS of (8. First, the functions sech x are exanded into the owers of e x. This results in a double sum of double integrals e (m+x (n+y cos x y dxdy, where m and n are nonnegative integers. The integral over y is easily calculated e (n+y cos x y To calculate the integral over x we need formula.64. from [7] ( dy = +. (m + + ix (m + ix e (n+x (m + ± ix dx = ( m+n e i 4 K ( i (m + (n +. Note that K (ix = (Y (x + ij (x, x R. As a result the double integral in (8 reduces to a combination of double sums ( Z (m + (n +, m,n= where Z is either Bessel J or Neumann Y function. Acknowledgements. The author of this aer wish to thank Dr. Lawrence Glasser for valuable corresondence and comments. G.H. Hardy, Note on the function x e (x t dt, Quart. J. Pure Al. Math., 5, (9. B. Cais, On the transformation of infinite series (unublished. B. Berndt and G.E. Andrews, Ramanujan s lost notebook, art IV, Sringer New York (5. 4 E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, nd.ed., Oxford University Press ( S. Ramanujan, Some definite integrals, Mess. Math., XLIV,  8 (95. 6 E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, Cambridge university ress ( I.S. Gradshteyn, and I.M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, Boston (. 8 S. Ramanujan, Some definite integrals, J. Indian Math. Soc., XI, 887 (99. 9 M.L. Glasser, Generalization of a definite integral of Ramanujan, J. Indian Math. Soc., 7, 5 (97. M.L. Glasser, A Remarkable Definite Integral, arxiv:8.66v (. J.M. Borwein, M.L. Glasser, R.C. McPhedran, J.G. Wan, I.J. Zucker, Lattice sums then and now, Cambridge University Press (.