References [1] M. Google Scholar [2] A. Google Scholar [3] M. Google Scholar [4] T. Google Scholar [7] M. Google Scholar [8] D. Google Scholar [9] G. Google Scholar [10] L. Google Scholar [11] D. Google Scholar [14] M. Google Scholar [18] B. Miller View author publications. View author publications. It may takes up to minutes before you received it. Please note : you need to verify every book you want to send to your Kindle. Check your mailbox for the verification email from Amazon Kindle.
Related Booklists. This the only of the versions in Z-Library of this very useful book that has the real cover. A great book for the student, engineer, professors and mathematicians. Let 1. Inversion If f t is absolutely integrable on [0, R] for every finite R, and the integral 1. Table 1. For a generalization of 1. For 1. For mathematical interpretations of 1.
These sources supplement the references that are quoted in the text. For the Riemann— Lebesgue lemma see Olver b, p. For 2 1. For a proof of the Jordan Curve Theorem see, for example, Dienes , pp. The theorem is valid with less restrictive conditions than those assumed here. For the operations on series, see Henrici , Chapter 1 or Olver b, pp. See also Andrews et al. The Extended Inversion Theorem is proved in a similar way. For the Horner scheme, see Burnside and Panton , pp.
The double Horner scheme is derived similarly. Chapter 2 Asymptotic Approximations F. Olver1 and R. Wong2 Areas 2. Integrals of a Real Variable.
Contour Integrals. Mellin Transform Methods. Distributional Methods. Differential Equations with a Parameter Difference Equations. Sums and Sequences. Copyright 2 Liu 41 42 Asymptotic Approximations Areas 2. Symbolically, 2. Condition 2. If c is a finite limit point of X, then 2. The symbols o and O can be used generically. For example, 2.
For the more general integral 2. These references and Wong , Chapter 2 also contain examples. If p b is finite, then both endpoints contribute: 2. However, cancellation does not take place near the endpoints, owing to lack of symmetry, nor in the neighborhoods of zeros of p0 t because p t changes relatively slowly at these stationary points.
Assume that q t again has the expansion 2. For proofs of the results of this subsection, error bounds, and an example, see Olver For other estimates of the error term see Lyness For extensions to oscillatory integrals with logarithmic singularities see Wong and Lin Thus 2. The integral 2. I z converges at b absolutely and uniformly with respect to z. Thus the right-hand side of 2. In consequence, the asymptotic expansion obtained from 2.
Then 7. For these and other error bounds see Olver b, pp. Note that some of these re-expansions themselves involve the complementary error function. For further inequalities of these types see Qi and Mei Equalities in 8. Next, define Z x 8. Also, define 8. For bounds on Rn a, z when a is real and z is complex see Olver b, pp.
Also, 8. The expansion 8. See Tricomi b for these approximations, together with higher terms and extensions to complex variables. For related expansions involving Hermite polynomials see Pagurova For the function en z defined by 8.
For 8. Special cases are given by 8. Then 8. For error bounds for 8. For a historical profile of Bx a, b see Dutka The functions Fk are defined by 8. See also Cuyt et al. Other Integral Representations 8. Here 8. Integral representations of Mellin—Barnes type for Ep z follow immediately from 8. For an extensive treatment of E1 z see Chapter 6.
In Figures 8. See p. Principal value. There is a branch cut along the negative real axis. Figure 8. Where the sectors of validity of 8. In particular, 8. For the functions on the right-hand sides of 8. From here on it is assumed that unless indicated otherwise the functions si a, z , ci a, z , Si a, z , and Ci a, z have their principal values.
I1 , I2 , I3 are depicted in Figure 9. Paths I1 , I2 , I3. These properties include Wronskians, asymptotic expansions, and information on zeros. For further generalizations via integral representations see Chin and Hedstrom , Janson et al. Each of the functions Ak z, p and Bk z, p satisfies the differential equation 9.
For information, including asymptotic approximations, computation, and applications, see Levey and Felsen , Constantinides and Marhefka , and Michaeli Applications 9. The function Ai x first appears as an integral in two articles by G. Airy on the intensity of light in the neighborhood of a caustic Airy , Details of the Airy theory are given in van de Hulst in the chapter on the optics of a raindrop.
See also Berry , The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic WKBJ solutions are exponential on one side and oscillatory on the other.
The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood. Within classical physics, they appear prominently in physical optics, electromagnetism, radiative transfer, fluid mechanics, and nonlinear wave propagation. Examples dealing with the propagation of light and with radiation of electromagnetic waves are given in Landau and Lifshitz Extensive use is made of Airy functions in investigations in the theory of electromagnetic diffraction and radiowave propagation Fock A quite different application is made in the study of the diffraction of sound pulses by a circular cylinder Friedlander In fluid dynamics, Airy functions enter several topics.
In the study of the stability of a twodimensional viscous fluid, the flow is governed by the Orr—Sommerfeld equation a fourth-order differential equation. Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads after choosing solvable equations with similar asymptotic properties to Airy functions.
Other applications appear in the study of instability of Couette flow of an inviscid fluid. These examples of transitions to turbulence are presented in detail in Drazin and Reid with the problem of hydrodynamic stability. The investigation of the transition between subsonic and supersonic of a two-dimensional gas flow leads to the Euler—Tricomi equation Landau and Lifshitz An application of Airy functions to the solution of this equation is given in Gramtcheff Airy functions play a prominent role in problems defined by nonlinear wave equations.
These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg— de Vries KdV equation a third-order nonlinear partial differential equation. The KdV equation and solitons have applications in many branches of physics, including plasma physics lattice dynamics, and quantum mechan- ics. Ablowitz and Segur , Ablowitz and Clarkson , and Whitham This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point.
A study of the semiclassical description of quantum-mechanical scattering is given in Ford and Wheeler a,b. In the case of the rainbow, the scattering amplitude is expressed in terms of Ai x , the analysis being similar to that given originally by Airy for the corresponding problem in optics.
An application of the Scorer functions is to the problem of the uniform loading of infinite plates Rothman a,b. Computation 9. Since these expansions diverge, the accuracy they yield is limited by the magnitude of z. For details, including the application of a generalized form of Gaussian quadrature, see Gordon , Appendix A and Schulten et al. Gil et al.
In the first method the integration path for the contour integral 9. For the second method see also Gautschi a. The methods for Ai0 z are similar. For quadrature methods for Scorer functions see Gil et al. See also Fabijonas et al. For the computation of the zeros of the Scorer functions and their derivatives see Gil et al.
Airy and Related Functions 9. Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun , Chapter 10 , together with some auxiliary functions for large arguments.
Precision is 10D. Precision is 7S. In the expansions Then The bounds Corresponding error bounds for Then in For error bounds for the first of For proofs and also for the corresponding expansions for second derivatives see Olver For higher coefficients in In this way there is less usage of many-valued functions. Corresponding points of the mapping are shown in Figures They are given parametrically by For proofs of the above results and for error bounds and extensions of the regions of validity see Olver b, pp.
For further results see Dunster a , Wang and Wong , and Paris Figure E1 and E2 are the points Figure Domain K unshaded. For sign properties of the forward differences that are defined by For further monotonicity properties see Elbert , Lorch , , , Lorch and Szego , , and Muldoon For inequalities for zeros arising from monotonicity properties see Laforgia and Muldoon For the next three terms in See also Laforgia In particular, with the notation as below, For error bounds for See also Spigler For the first zeros rounded numerical values of the coefficients are given by For derivations and further information, including extensions to uniform asymptotic expansions, see Olver , The latter reference includes numerical tables of the first few coefficients in the uniform asymptotic expansions.
Figures The zeros of Yn0 nz have a similar pattern to those of Yn nz. For further information, including uniform asymptotic expansions, extensions to other branches of the functions and their derivatives, and extensions to half-integer values of the order, see Olver There is an inaccuracy in Figures 11 and 14 in this reference.
Each curve that represents an infinite string of nonreal zeros should be located on the opposite side of its straight line asymptote. This inaccuracy was repeated in Abramowitz and Stegun , Figures 9.
See Kerimov and Skorokhodov a,b and Figures See also Cruz and Sesma ; Cruz et al. The zeros of Hn nz have a similar pattern to 1 2 2 0 those of Hn nz. The zeros of Hn nz and Hn nz 1 are the complex conjugates of the zeros of Hn nz and 1 0 Hn nz , respectively. In consequence of Also, in consequence of Also, Differential equations for products can be obtained from In particular, use of For these and further results see Miller , pp.
The general terms in For the functions 0 F1 and F see For the error term in It needs to be noted that the results See also Kerimov and Skorokhodov b,a. For further properties of the Bickley function, including asymptotic expansions and generalizations, see Amos , and Luke , Chapter 8. Equations In place of However, care needs to be exercised with the branches of the phases. All fractional powers take their principal values.
Accuracy in For the next six terms in the series Examples Z For direct and inverse Laplace transforms of Kelvin functions see Prudnikov et al. Applications These asymptotic expansions are uniform with respect to z, including cut neighborhoods of z0 , and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation.
This equation governs problems in acoustic and electromagnetic wave propagation. It is fundamental in the study of electromagnetic wave transmission. Consequently, Bessel functions Jn x , and modified Bessel functions In x , are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. Bessel functions enter in the study of the scattering of light and other electromagnetic radiation, not only from cylindrical surfaces but also in the statistical analysis involved in scattering from rough surfaces.
More recently, Bessel functions appear in the inverse problem in wave propagation, with applications in medicine, astronomy, and acoustic imaging. On separation of variables into cylindrical coordinates, the Bessel functions Jn x , and modified Bessel functions In x and Kn x , all appear. Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation.
The McLachlan reference also includes other applications of Kelvin functions. Computation In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. Temme shows how to overcome this difficulty by use of the Maclaurin expansions for these coefficients or by use of auxiliary functions. Similar observations apply to the computation of modified Bessel functions, spherical Bessel functions, and Kelvin functions.
And since there are no error terms they could, in theory, be used for all values of z; however, there may be severe cancellation when z is not large compared with n2. Then Jn x and Yn x can be generated by either forward or backward recurrence on n when n x then to maintain stability Jn x has to be generated by backward recurrence on n, and Yn x has to be generated by forward recurrence on n.
Necessary values of the first derivatives of the functions are obtained by the use of See also Segura , Real Zeros See Olver , pp. Multiple Zeros See Kerimov and Skorokhodov c, , , See Lehman et al. Kontorovich—Lebedev Transform See Ehrenmark Products For infinite integrals involving products of two Bessel functions of the first kind, see Linz and Kropp , Gabutti , Ikonomou et al. Only a few of the more comprehensive of these early tables are included in the listings in the following subsections.
Also, for additional listings of tables pertaining to complex arguments see Babushkina et al. Also included are auxiliary functions to facilitate interpolation of the tables of Y0 x , Y1 x for small values of x, as well as auxiliary functions to compute all four functions for large values of x.
Other zeros of this function can be obtained by reflection in the imaginary axis. Also included are auxiliary functions to facilitate interpolation of the tables of K0 x , K1 x for small values of x. For the notation replace j, y, i, k by j, y, i 1 , k, respectively. Luke , Tables 9. Bickley Functions Blair et al. Spherical Bessel Functions Delic Kelvin Functions Luke , Table 9.
References General References The following list gives the references or other indications of proofs that were used in constructing the various sections of this chapter. Then for the cross-products apply For The condition 0 in See also Olver b, pp.
See also Watson , pp. To derive Similarly for The general term in Higher terms can be calculated via Similar methods can be used for See also Bickley et al. In the latter reference t in The zeros depicted in Figures To verify Then compare the result with the corresponding expansion of the right-hand side obtained from In the case of The verification of Some modifications of the proof of For the first result in See also For the cross-products apply Also use An error in the conditions has been corrected.
Both cases of For the statement concerning the accuracy of For some results it is necessary to use the connection formulas For Conditions b see Lebedev et al.
The Wronskian For the first of Similarly for the second of Next, apply Then refer to The version of Then from Software Images icon An illustration of two photographs. Images Donate icon An illustration of a heart shape Donate Ellipses icon An illustration of text ellipses. Handbook of mathematical functions with formulas, graphs, and mathematical tables Item Preview. EMBED for wordpress.
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