# The properties, inequalities and numerical approximation of modified bessel functions.

Abstract. Some new properties of kernels of modified
Kontorovitch-Lebedev integral transforms--modified Bessel functions of
the second kind with complex order [K.sub.1/2 + i[beta]] (x) are
presented. Inequalities giving estimations for these functions with
argument x and parameter [beta] are obtained. The polynomial approximations of these functions as a solutions of linear differential
equations with polynomial coefficients and their systems are proposed.

Key words. Chebyshev polynomials, modified Bessel functions, Lanczos Tau method, Kontorovich-Lebedev integral transforms

AMS subject classifications. 33C10, 33F05, 65D20

1. Some properties of the functions Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x). In this section new properties of the kernels of modified Kontorovitch-Lebedev integral transforms are deduced, and some of their known properties are collected, which are necessary later on.

It is possible to write the kernels of these transforms in the form

Re[K.sub.1/2 + i[beta]](x) = [K.sub.1/2 + i[beta]](x) + [K.sub.1/2 + i[beta]](x)/2 and Im[K.sub.1/2 + i[beta]](x) = [K.sub.1/2 + i[beta]](x) - [K.sub.1/2 + i[beta]](x)/2,

where [K.sub.v] (x) is the modified Bessel function of the second kind (also called MacDonald function).

The functions Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x) have integral representations [8]

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The vector-function ([y.sub.1](x), [y.sub.2](x)) with the components [y.sub.1](x) = Re[K.sub.1/2 + i[beta]](x), [y.sub.2](x) = Im[K.sub.1/2 + i[beta]](x) is the solution of the system of differential equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The functions Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x) are even and odd functions, respectively of the variable [beta],

Re[K.sub.1/2 + i[beta]](x) = Re[K.sub.1/2 + i[beta]](x),

Im[K.sub.1/2 + i[beta]](x) = -Im[K.sub.1/2 + i[beta]](x).

The functions Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x) are related to the modified Bessel functions of the first kind [I.sub.v](x) as follows,

(1.3) Re[K.sub.1/2 + i[beta]](x) = [pi]/cosh([pi][beta]) Re[I.sub.-1/2 - i[beta]](x) - Re[I.sub.1/2 + i[beta]] (x)/2,

(1.4) Im[K.sub.1/2 + i[beta]](x) = [pi]/cosh([pi][beta]) Im[I.sub.-1/2 - i[beta]] - Im[I.sub.1/2 + i[beta]](x)/2.

The expansion of [I.sub.1/2 + i[beta]](x) in ascending powers of x has the form

(1.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [a.sub.k] and [b.sub.k] satisfy the following recurrence relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The expansion of [I.sub.-1/2 - i[beta]](x) in ascending powers of x has the form

(1.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [c.sub.k] and [d.sub.k] satisfy the following recurrence relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The expansions (1.5) and (1.6) converge for all 0 < x < [infinity] and 0 [less than or equal to] [beta] < [infinity].

It follows from (1.1)-(1.2) that it is possible to write Re[K.sub.1/2 + i[beta]](x) in the form of the Fourier cosinus-transform

(1.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and Im[K.sub.1/2 + i[beta]](x) in the form of the Fourier sinus-transform

(1.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The inversion formulas have the respective forms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or, in integral form,

(1.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Differentiating equations (1.9) and (1.10) with respect to t, we obtain

(1.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It follows from (1.9) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and from (1.11) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Differentiating (1.9) and (1.10) 2n times with respect to t, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

from which there follows, for t = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Differentiating (1.9) and (1.10) 2n + 1 times with respect to t, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

whence, for t = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For the computation of certain integrals of the functions Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x), integral identities are useful. They reduce this problem to the computation of some other integrals of elementary functions.

PROPOSITION 1.1. If f is absolutely integrable on [0, [infinity]), then the following identities hold,

(1.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [F.sub.C] (t) is the Fourier cosinus-transform of f ([beta]), and [F.sub.S] (t) the Fourier sinus-transform of f ([beta]).

Proof. Multiplying both sides of the equalities (1.7) and (1.8) by f ([beta]), integrating with respect to [beta] from 0 to [infinity], and applying Fubini's theorem for singular integrals with parameter [22], we obtain (1.12) and (1.13).

PROPOSITION 1.2. If f is absolutely integrable on [0, [infinity]), then the following identities hold

(1.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof. This follows from (1.9)-(1.10) and from Fubini's theorem [22].

The equations (1.12)-(1.15) are useful for the simplification and the calculation of different integrals containing Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x).

For example, let f ([beta]) = [e.sup.-[alpha][beta]], then [F.sub.C] (t) = [square root of (2[pi] [alpha]/[[alpha].sup.2]+[t.sup.2])], [F.sub.S](t) = [square root of (2/[pi] t/[[alpha].sup.2] + [t.sup.2])] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If f ([beta]) = [GAMMA](1/4 + i[beta]/2)[GAMMA](1/4 - i[beta]/2), then [F.sub.C](t) = 2[pi]/[square root of (cosh t)] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If f ([beta]) = sinh(2[pi][beta])/cosh(2[pi][beta])+cos(2[pi][beta]), |Re[alpha]| < 1/2, then [F.sub.S](t) = cosh([alpha]t/[square root of (2[pi]sinh t/2)] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

REMARK 1. 3. All formulas of the present paragraph remain valid if x is changed to z lying in the right-hand half-plane.

1.1. The Laplace transform of Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x).

The Laplace transform of [K.sub.i[beta]](x) is computed in [21]. We use the representation (1.1) for the evaluation of the Laplace transformation of Re[K.sub.1/2 + i[beta]](x). We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Equivalently, we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For the evaluation of the Laplace transform of Im[K.sub.1/2 + i[beta]](x) we utilize the representation (1.2). We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or, equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We note that these equations can also be obtained directly from the formula for the Laplace transforms of [K.sub.v] (x) by separating real and imaginary parts.

1.2. The asymptotic behavior of Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x) for x [right arrow] 0, x [right arrow] [infinity] and [beta] [right arrow] [infinity]. For Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x) the following asymptotic formulas for [beta] [right arrow] [infinity] are valid [8],

Re[K.sub.1/2 + i[beta]](x) ~ [([pi]/x).sup.1/2] [e.sup.-[pi][beta]/2] cos ([beta]1n [beta] - [beta] - [beta]1n x/2),

Im[K.sub.1/2 + i[beta]](x) ~ [([pi]/x).sup.1/2] [e.sup.-[pi][beta]/2] sin ([beta]1n [beta] - [beta] - [beta]1n x/2),

where x is a fixed positive number.

It follows immediately from (1.3)-(1.6) that for x [right arrow] 0 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

whence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For large values x the following asymptotic expansion is valid [9]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

(v, k) - (4[v.sup.2] - [1.sup.2])(4[v.sup.2] - [3.sup.2]) ... (4[v.sup.2] - [(2k - 1).sup.2])/[2.sup.2k]k!.

In particular, therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

1.3. The series expansions in powers of [beta]. The solutions of problems in mathematical physics connected with the use of the Kontorovitch-Lebedev integral transforms are often expressed as integrals with respect to [beta] of the functions [K.sub.i[beta]](x), Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x). Both the asymptotic expansions of these integrals for large values [beta], and the expansions of these functions in powers of [beta], are of interest for the analysis of the behavior of these integrals.

The expansions of these functions in powers of [beta] are deduced from their integral representations (1.1)-(1.2). Substituting in them cos([beta]t) and sin([beta]t) by their series expansions and interchanging the order of the summation and integration, we obtain

(1.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

These functions are entire functions in the variable [beta], and therefore the series converge for all real values of [beta]. From these expansions it is possible to obtain the series for the derivatives and for the integrals of these functions with respect to the variable [beta], which will converge for all real [beta] also. Similar integrals for the spherical functions are stated in [23].

It's possible to rewrite the expansions (1.16)-(1.18) in terms of Laplace transforms as follows,

(1.19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This form of writing may be more convenient since it is possible to use numerical methods for evaluating Laplace transforms.

The expansions (1.19)-(1.21) are convenient for the calculation of the kernels of Kontorovitch-Lebedev integral transforms for small values [beta].

2. Inequalities for the MacDonald functions [K.sub.i[beta]](x), Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x). It follows from (1.1) that for all [beta] E [0, [infinity])

|Re[K.sub.1/2 + i[beta]](x)| [less than or equal to] [K.sub.1/2](x) = [([pi]/2x).sup.1/2][e.sup.-x],

and it follows from (1.2) that for all [beta] [member of] [0, [infinity])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where B is some positive constant [10],[11].

In [4], for arbitrary v = [sigma] + i[beta], [sigma] [greater than or equal to] 0, the following inequality is derived

|[I.sub.v](x)| [less than or equal to] [e.sup.[pi]|[beta]|/2] [I.sub.[sigma]],(x).

Taking advantage of the formula [4]

|[K.sub.v](x)| [less than or equal to] ([C.sub.1] (x, [sigma]) + [C.sub.2] (x, [sigma])[|[beta]|.sup.[sigma] - 1/2]) [e.sup.-[pi]|[beta]|/2],

we obtain that beginning with some T, |[beta]| > T,

|[K.sub.1/2 + i[beta]](x)| [less than or equal to] C(x)[e.sup.-[pi]|[[beta]|/2].

But this inequality is too rough and may be insufficient for conducting various proofs. To obtain more refined inequalities, we use [5]

(2.1) |[K.sub.i[beta]](x)| [less than or equal to] [Ax.sup.-1/4][e.sup.-[pi]|[beta]|/2],

where A is some positive constant, and the representations [8]

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

LEMMA 2.1. The following inequalities hold for x > 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [c.sub.0] and c are some positive constants.

Proof. We estimate the second additive term in (2.2), using the inequality (2.1),

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We next estimate the third additive term,

(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Combining the first term and estimates (2.3) and (2.4), we obtain the required inequality.

Furthermore, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For future use, an analysis of the behavior of the modified Bessel function [K.sub.[sigma] + i[beta]](x) for large values of [beta] is necessary.

LEMMA 2.2. For 0 [less than or equal to] [sigma] [less than or equal to] 1/2, |[beta]| [greater than or equal to] [[beta].sub.0] [greater than or equal to] 1, x [greater than or equal to] [x.sub.0] [greater than or equal to] 1, the following inequality holds,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [c.sub.1] > 0, [c.sub.2] > 0, [c.sub.1], [c.sub.2], [[beta].sub.0], [x.sub.0] are some constants.

Proof. We use the formula [6]

[K.sub.[micro]](x) = [pi]/2 sin([pi][micro]) ([I.sub.-[micro]](x) - [I.sub.[micro]] (x)), [micro], = [sigma] + i[beta].

1. We first estimate sin([pi][micro]). It is possible to show that for |[beta]| [greater than or equal to] [[beta].sup.(1).sub.0] > 0, [[beta].sup.(1).sub.0] some constant, the following inequality is valid

(2.5) [a.sub.1][e.sup.[pi]|[beta]|] [less than or equal to] |sin([pi][micro])| [less than or equal to]< [a.sub.2][e.sup.[pi]|[beta]|],

where [a.sub.1] > 0, [a.sub.2] > 0, [a.sub.1], [a.sub.2] are some constants.

2. We next estimate [I.sub.[micro]](x), [micro], = [sigma] + i[beta], [sigma] [greater than or equal to] 0. The following inequality is derived for [sigma] [greater than or equal to] 0 in [4]

|[I.sub.[micro]](x)| [less than or equal to] [I.sub.o](x) [(x/2).sup.[sigma]][e.sup.[pi]|[beta]|/2]/[GAMMA]([sigma] + 1).

It follows from the asymptotics of [I.sub.0](x) [10] for large x that for x [greater than or equal to] [x.sup.(1).sub.0] > 0, [x.sup.(1).sub.0] some constant,

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [a.sub.3] > 0, [a.sub.3] some constant.

3. We finally estimate [I.sub.-[micro]], (x), [micro], = [sigma] + i[beta], [sigma] [greater than or equal to] 0. Proceeding analogously [4], we can rewrite [I.sub.-[micro]](x) in the form

[I.sub.-[micro]](x) = [(x/2).sup.-[micro]]/[GAMMA](1- [micro]) [psi](x, [micro]),

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then |k - [micro]|= [square root of ([(k - [sigma]).sup.2] + [[beta].sup.2])] [greater than or equal to] k - 1 for 0 [less than or equal to] [sigma] [less than or equal to] 1/2, k = 2,3, ... , and [square root of ([(1 - [sigma]).sup.2] + [[beta].sup.2])] [greater than or equal to] 1/2, [beta] arbitrary. Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. We obtain, after some calculations, that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using for |[beta]| [greater than or equal to] [[beta].sup.(2).sub.0]) [greater than or equal to] 1, 0 [less than or equal to] [sigma] [less than or equal to] 1/2, the expansion of the gamma-function from [5] and the asymptotics [6] for [I.sub.1](x) we obtain that beginning with some [x.sup.(2).sub.0], x [greater than or equal to] [x.sup.(2).sub.0] [greater than or equal to] 1, the following estimation holds,

(2.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[a.sub.4] > 0, [a.sub.4] some constant.

Combining the estimations (2.5)-(2.7), we obtain that for 0 [less than or equal to] [sigma] [less than or equal to] 1/2, |[beta]| [greater than or equal to] [[beta].sub.0] [greater than or equal to] 1, x [greater than or equal to] [x.sub.0] [greater than or equal to] 1, [[beta].sub.0] = max([[beta].sup.(1).sub.0], [[beta].sup.(2).sub.0], [x.sub.0] = max([x.sup.(1).sub.0], [x.sup.(2).sub.0])) the following inequality is valid,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Denoting [c.sub.1] = [pi]/2[a.sub.1] [a.sub.3], [c.sub.2] = [pi]/2[a.sub.1] [a.sub.4], we obtain the statement of the lemma.

The properties of the modified Kontorovitch-Lebedev integral transforms are considered in [7]-[15].

3. Tau method approximation for modified Bessel function of imaginary order. Several approaches for the evaluation of the modified Bessel functions are elaborated in [1]-[2]. The Tau method [3] realization, with minimal residue choice for the determination of the polynomial approximations of the solutions of the second order differential equations with polynomial coefficients [16] of the following form

([a.sub.0][y.sup.2] + [a.sub.5]y)v"(y) + ([a.sub.1]y + [a.sub.2])v'(y) + [a.sub.3]v(y) = 0, v(0) = [a.sub.4], y [member of] [0,1],

is supposed. An n-th approximation of the solution is sought in the form of the n-th degree polynomial [v.sub.n](y), which is the solution of the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the coefficients [a.sub.i], i = 0, ... , 5, may be expressed by coefficients [b.sub.i], i = 0, ... , 5, [a.sub.n+2] = [sin.sup.2] [pi]/4(n+2)--the leftmost root of the shifted Chebyshev polynomial of the n + 2-th degree [T.sub.n+2]* (y) in the interval [0 ,1], [[tau].sub.n+2]--undefined coefficient.

The problem about determination of the polynomial [P.sub.n](y) = [n.summation over (k=0)] [p.sub.k.][y.sup.k], which is the least deviated from zero on the interval [0,1] among all n-th degree polynomials, satisfying the pair of linear correlations on the coefficients [p.sub.0] = 0, [n.summation over (i=1)][c.sup.(n).sub.i][p.sub.i] = 1 was considered. The following theorem is proved [16]:

THEOREM 3.1. If the sequence of numbers [c.sup.(n).sub.i], i = 1, ... , n, is alternating, then the polynomial [[tau].sub.n][T.sub.n]* [(1 - [[alpha].sub.n])y + [[alpha].sub.n]] is the polynomial least deviating from zero in the uniform metric on [0,1] among all polynomials of degree n, satisfying the indicated pair of linear relations.

On the basis of this theorem it's shown (as suggested by us) in the Tau method residue, in a number of significant cases, is a minimal in the uniform metric on [0,1], among all possible polynomial residues permitting the Volterra integral equations solution.

We have the following differential equation with polynomial coefficients for the approximation and computing of the second kind modified Bessel function [K.sub.i[beta]](x):

[y.sup.2]v"(y) + 2(y + 1)v'(y) + (1/4 + [[beta].sup.2])v(y) = 0,

v(0) = 1,

and the Volterra integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We obtain the following recurrence formulas for the coefficients of canonical polynomials [Q.sub.m](y) = [m.summation over (k=0)][q.sub.km][y.sup.k] in this case:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The minimality of the residue suggested by us follows from the Theorem 3.1 as qom/|qom| = [(-1).sup.m], m = 0,1, ...

The advantages of this modification, as compared with usual and other tau-methods, is shown.

4. Tau method approximation for modified Bessel function of complex order. A new numerical scheme of the Tau method application is proposed for the solution of the second order linear differential equations systems, with the second order polynomial coefficients of the following kind:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

in the unknown vector-function v(y) = ([v.sub.i] (y), ... , [v.sub.k](y)). It is assumed to have only one solution. Integrating twice and carrying an addition in the right part in the kind of the vector-polynomial [P.sub.n] (y), we derive for the determination of the n-th approximation of the solution v(y) = ([v.sub.1](y), ... , [v.sub.k] (y)) the system of Volterra integral equations with polynomial kernels

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the coefficients [b.sup.(j).sub.i] and [a.sup.(j).sub.i], i = 0, ... , 3k + 2 and j = 1, ... , k, are connected in a definite way and [P.sub.jn+2] (y), j = 1, ... , k, are n + 2-th degree polynomials. The different variables of the vector residue choice and its minimization are analyzed. The recurrence formulas for the canonical vector-polynomials coefficients convenient for the calculations are given.

Consider the system of two second order differential equations (k = 2) in more detail. This case is of particular interest for differential equations with complex coefficients. The scheme of the integral form of the Tau Method described in this paper can be used for deriving polynomial approximations of hypergeometric and confluent hypergeometric functions of the first kind with complex parameters.

The modified Kontorovich-Lebedev integral transforms [7] with kernels Re[K.sub.1/2 + i[beta]] (x) = [K.sub.1/2 + i[beta]] (x) + [K.sub.1/2 - i[beta]](x)/2 and Im[K.sub.1/2 + i[beta]](x) = [K.sub.1/2 + i[beta]](x) - [K.sub.1/2 - i[beta]],(x)/2i, where [ is MacDonald's function, is of great importance in solving some problems of mathematical physics, in particular mixed boundary value problems for the HELMHOLTz equation in wedge and cone domains. We find it necessary to compute Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x) to use this transform in practice [13]. These functions also occur in solving some classes of dual integral equations with kernels which contain MacDonald's function of imaginary index [K.sub.i[beta]](x) [7]. Therefore, now we consider the second kind modified Bessel function [K.sub.1/2 + i[beta]](x) in more detail.

We have a system of two second order differential equations

[y.sup.2][v.sub.1]" + 2(y + 1)[v.sub.1]' + [[beta].sup.2][v.sub.1] + [beta][v.sub.2] = 0,

[y.sup.2][v.sub.2]" + 2(y + 1)[v.sub.2]' - [beta][v.sub.1] + [[beta].sup.2][v.sub.2] = 0,

[V.sub.1](0) = 1, [V.sub.2](0) = 0,

or the system of Volterra integral equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The following formulas for the coefficients of canonical vector-polynomials are derived [16]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

By means of computations is shown that the choice of the residue in the form [P.sub.jn+2](y) = [[tau].sub.jn+2][T.sub.n+2][(1 - [[alpha].sub.n+2])y + [[alpha]sub.n+2]], j = 1, 2, is optimal as compared with other known variants in this case too.

The applications for the numerical solution of boundary value problems in wedge domains are given in [18],[19].

Acknowledgments. The author wishes to thank the organizers of the Constructive Functions-Tech04 Conference for their encouragement and support.

Dedicated to Ed Saff on the occasion of his 60th birthday

* Received April 28, 2005. Accepted for publication December 22, 2005. Recommended by D. Lubinsky. The author's participation in the Constructive Functions-Tech04 Conference has been partially supported by the Russian Foundation of Basic Research, Grant 04-01-10790 and National Science Foundation, Grant DMS 0411729.

REFERENCES

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[3] J. H. FREILICH AND E. L. ORTIZ, Numerical solution of systems of ordinary differential equations with the Tau Method, Math. Comp., 39 (1982), pp. 467-479.

[4] A. G. GRINBERG, The Selected Questions of the Mathematical Theory of the Electric and Magnetic Phenomenons, (in Russian), Publishing House of AN SSSR, Moscow-Leningrad, 1948.

[5] N. N. LEBEDEV, Some Integral Transforms of the Mathematical Physics, PhD Thesis, (in Russian), Ioffe Physics-Technical Institute, Leningrad, 1950.

[6] N. N. LEBEDEV, The Special Functions and Their Applications, 2nd edition, revised and complemented, (in Russian), Fizmatgiz, Moscow-Leningrad, 1963.

[7] N. N. LEBEDEV AND I. P. SKALSKAYA, The dual integral equations connected with the Kontorovitch-Lebedev transform, (in Russian), Prikl. Matem. and Meehan., 38 (1974), pp. 1090-1097.

[8] N. N. LEBEDEV AND I. P. SKALSKAYA, Some integral transforms related to Kontorovitch-Lebedev transforms, (in Russian), in The Questions of the Mathematical Physics, pp. 68-79, Nauka, Leningrad, 1976.

[9] A. F. NIKIFOROV AND V. B. UVAROV, The Special Functions of the Mathematical Physics, (in Russian), Nauka, Moscow, 1975.

[10] A. P. PRUDNIKOV, Y. A. BRYCHKOV, AND O.I. MARICHEV, The Integrals and Series, (in Russian), Nauka, Moscow, 1981.

[11] A. P. PRUDNIKOV, Y. A. BRYCHKOV, AND O.I. MARICHEV, The Integrals and Series, Special Functions, (in Russian), Nauka, Moscow, 1983.

[12] V. B. PORUCHIKOV AND J.M. RAPPOPORT, Inversion formulas for modified Kontorovitch-Lebedev transforms, (in Russian), Diff. Uravn., 20 (1984), pp. 542-546.

[13] J. M. RAPPOPORT, Tables of modified Bessel Junctions [K.sub.1/2 + i[beta]](x), (in Russian), Nauka, Moscow, 1979.

[14] J. M. RAPPOPORT, Some properties of modified Kontorovitch-Lebedev integral transforms, (in Russian), Diff. Uravn., 21 (1985), pp. 724-727.

[15] J. M. RAPPOPORT, Some results for modified Kontorovitch-Lebedev integral transforms, in Proceedings of the 7th International Colloquium on Finite or Infinite Dimensional Complex Analysis, Marcel Dekker, 2000, pp. 473-477.

[16] J. M. RAPPOPORT, The canonical vector-polynomials at computation of the Bessel functions of the complex order, Comput. Math. Appl., 41 (2001), pp. 399-406.

[17] J. M. RAPPOPORT, Some numerical quadrature algorithms for the computation of the MacDonald Junction, in Proceedings of the Third ISAAC Congress, Progress in Analysis, vol. 2, World Scientific Publishing, 2003, pp. 1223-1230.

[18] J. M. RAPPOPORT, Tau method and numerical solution of some mixed boundary value problems, in Proceedings of the Conference on Computational and Mathematical Methods on Science and Engineering CMMSE-2004, Angstrom Laboratory, 2004, pp. 236-242.

[19] J. M. RAPPOPORT, Dual integral equations for some mixed boundary value problems, in Advances in Analysis, Proceedings of the 4th ISAAC Congress, World Scientific Publishing, 2005, pp. 167-176,

[20] A. GIL, J. SEGURA AND N. M. TEMME, Evaluation of the modified Bessel function of the third kind of imaginary orders, J. Comput. Phys., 175 (2002), pp. 398-411.

[21] 1. N. SNEDDON, The Use of Integral Transforms, McGraw-Hill, New York, 1972.

[22] N. VIENER, Fourier Integral and Some Its Applications, (in Russian), Fizmatgiz, Moscow, 1963.

[23] M. 1. ZHURINA AND L. N. KARMAZINA, The Tables and Formulas for the Spherical Functions [P.sup.m.sub.-1/2 + i[tau]](z), (in Russian), VC AN SSSR, Moscow, 1962.

JURI M. RAPPOPORT Russian Academy of Sciences, Vlasov Street, Building 27, Apt. 8, Moscow 117335, Russia (jmrap@1andau.ac.ru).

Key words. Chebyshev polynomials, modified Bessel functions, Lanczos Tau method, Kontorovich-Lebedev integral transforms

AMS subject classifications. 33C10, 33F05, 65D20

1. Some properties of the functions Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x). In this section new properties of the kernels of modified Kontorovitch-Lebedev integral transforms are deduced, and some of their known properties are collected, which are necessary later on.

It is possible to write the kernels of these transforms in the form

Re[K.sub.1/2 + i[beta]](x) = [K.sub.1/2 + i[beta]](x) + [K.sub.1/2 + i[beta]](x)/2 and Im[K.sub.1/2 + i[beta]](x) = [K.sub.1/2 + i[beta]](x) - [K.sub.1/2 + i[beta]](x)/2,

where [K.sub.v] (x) is the modified Bessel function of the second kind (also called MacDonald function).

The functions Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x) have integral representations [8]

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The vector-function ([y.sub.1](x), [y.sub.2](x)) with the components [y.sub.1](x) = Re[K.sub.1/2 + i[beta]](x), [y.sub.2](x) = Im[K.sub.1/2 + i[beta]](x) is the solution of the system of differential equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The functions Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x) are even and odd functions, respectively of the variable [beta],

Re[K.sub.1/2 + i[beta]](x) = Re[K.sub.1/2 + i[beta]](x),

Im[K.sub.1/2 + i[beta]](x) = -Im[K.sub.1/2 + i[beta]](x).

The functions Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x) are related to the modified Bessel functions of the first kind [I.sub.v](x) as follows,

(1.3) Re[K.sub.1/2 + i[beta]](x) = [pi]/cosh([pi][beta]) Re[I.sub.-1/2 - i[beta]](x) - Re[I.sub.1/2 + i[beta]] (x)/2,

(1.4) Im[K.sub.1/2 + i[beta]](x) = [pi]/cosh([pi][beta]) Im[I.sub.-1/2 - i[beta]] - Im[I.sub.1/2 + i[beta]](x)/2.

The expansion of [I.sub.1/2 + i[beta]](x) in ascending powers of x has the form

(1.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [a.sub.k] and [b.sub.k] satisfy the following recurrence relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The expansion of [I.sub.-1/2 - i[beta]](x) in ascending powers of x has the form

(1.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [c.sub.k] and [d.sub.k] satisfy the following recurrence relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The expansions (1.5) and (1.6) converge for all 0 < x < [infinity] and 0 [less than or equal to] [beta] < [infinity].

It follows from (1.1)-(1.2) that it is possible to write Re[K.sub.1/2 + i[beta]](x) in the form of the Fourier cosinus-transform

(1.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and Im[K.sub.1/2 + i[beta]](x) in the form of the Fourier sinus-transform

(1.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The inversion formulas have the respective forms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or, in integral form,

(1.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Differentiating equations (1.9) and (1.10) with respect to t, we obtain

(1.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It follows from (1.9) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and from (1.11) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Differentiating (1.9) and (1.10) 2n times with respect to t, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

from which there follows, for t = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Differentiating (1.9) and (1.10) 2n + 1 times with respect to t, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

whence, for t = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For the computation of certain integrals of the functions Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x), integral identities are useful. They reduce this problem to the computation of some other integrals of elementary functions.

PROPOSITION 1.1. If f is absolutely integrable on [0, [infinity]), then the following identities hold,

(1.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [F.sub.C] (t) is the Fourier cosinus-transform of f ([beta]), and [F.sub.S] (t) the Fourier sinus-transform of f ([beta]).

Proof. Multiplying both sides of the equalities (1.7) and (1.8) by f ([beta]), integrating with respect to [beta] from 0 to [infinity], and applying Fubini's theorem for singular integrals with parameter [22], we obtain (1.12) and (1.13).

PROPOSITION 1.2. If f is absolutely integrable on [0, [infinity]), then the following identities hold

(1.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof. This follows from (1.9)-(1.10) and from Fubini's theorem [22].

The equations (1.12)-(1.15) are useful for the simplification and the calculation of different integrals containing Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x).

For example, let f ([beta]) = [e.sup.-[alpha][beta]], then [F.sub.C] (t) = [square root of (2[pi] [alpha]/[[alpha].sup.2]+[t.sup.2])], [F.sub.S](t) = [square root of (2/[pi] t/[[alpha].sup.2] + [t.sup.2])] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If f ([beta]) = [GAMMA](1/4 + i[beta]/2)[GAMMA](1/4 - i[beta]/2), then [F.sub.C](t) = 2[pi]/[square root of (cosh t)] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If f ([beta]) = sinh(2[pi][beta])/cosh(2[pi][beta])+cos(2[pi][beta]), |Re[alpha]| < 1/2, then [F.sub.S](t) = cosh([alpha]t/[square root of (2[pi]sinh t/2)] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

REMARK 1. 3. All formulas of the present paragraph remain valid if x is changed to z lying in the right-hand half-plane.

1.1. The Laplace transform of Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x).

The Laplace transform of [K.sub.i[beta]](x) is computed in [21]. We use the representation (1.1) for the evaluation of the Laplace transformation of Re[K.sub.1/2 + i[beta]](x). We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Equivalently, we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For the evaluation of the Laplace transform of Im[K.sub.1/2 + i[beta]](x) we utilize the representation (1.2). We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or, equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We note that these equations can also be obtained directly from the formula for the Laplace transforms of [K.sub.v] (x) by separating real and imaginary parts.

1.2. The asymptotic behavior of Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x) for x [right arrow] 0, x [right arrow] [infinity] and [beta] [right arrow] [infinity]. For Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x) the following asymptotic formulas for [beta] [right arrow] [infinity] are valid [8],

Re[K.sub.1/2 + i[beta]](x) ~ [([pi]/x).sup.1/2] [e.sup.-[pi][beta]/2] cos ([beta]1n [beta] - [beta] - [beta]1n x/2),

Im[K.sub.1/2 + i[beta]](x) ~ [([pi]/x).sup.1/2] [e.sup.-[pi][beta]/2] sin ([beta]1n [beta] - [beta] - [beta]1n x/2),

where x is a fixed positive number.

It follows immediately from (1.3)-(1.6) that for x [right arrow] 0 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

whence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For large values x the following asymptotic expansion is valid [9]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

(v, k) - (4[v.sup.2] - [1.sup.2])(4[v.sup.2] - [3.sup.2]) ... (4[v.sup.2] - [(2k - 1).sup.2])/[2.sup.2k]k!.

In particular, therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

1.3. The series expansions in powers of [beta]. The solutions of problems in mathematical physics connected with the use of the Kontorovitch-Lebedev integral transforms are often expressed as integrals with respect to [beta] of the functions [K.sub.i[beta]](x), Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x). Both the asymptotic expansions of these integrals for large values [beta], and the expansions of these functions in powers of [beta], are of interest for the analysis of the behavior of these integrals.

The expansions of these functions in powers of [beta] are deduced from their integral representations (1.1)-(1.2). Substituting in them cos([beta]t) and sin([beta]t) by their series expansions and interchanging the order of the summation and integration, we obtain

(1.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

These functions are entire functions in the variable [beta], and therefore the series converge for all real values of [beta]. From these expansions it is possible to obtain the series for the derivatives and for the integrals of these functions with respect to the variable [beta], which will converge for all real [beta] also. Similar integrals for the spherical functions are stated in [23].

It's possible to rewrite the expansions (1.16)-(1.18) in terms of Laplace transforms as follows,

(1.19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(1.21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This form of writing may be more convenient since it is possible to use numerical methods for evaluating Laplace transforms.

The expansions (1.19)-(1.21) are convenient for the calculation of the kernels of Kontorovitch-Lebedev integral transforms for small values [beta].

2. Inequalities for the MacDonald functions [K.sub.i[beta]](x), Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x). It follows from (1.1) that for all [beta] E [0, [infinity])

|Re[K.sub.1/2 + i[beta]](x)| [less than or equal to] [K.sub.1/2](x) = [([pi]/2x).sup.1/2][e.sup.-x],

and it follows from (1.2) that for all [beta] [member of] [0, [infinity])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where B is some positive constant [10],[11].

In [4], for arbitrary v = [sigma] + i[beta], [sigma] [greater than or equal to] 0, the following inequality is derived

|[I.sub.v](x)| [less than or equal to] [e.sup.[pi]|[beta]|/2] [I.sub.[sigma]],(x).

Taking advantage of the formula [4]

|[K.sub.v](x)| [less than or equal to] ([C.sub.1] (x, [sigma]) + [C.sub.2] (x, [sigma])[|[beta]|.sup.[sigma] - 1/2]) [e.sup.-[pi]|[beta]|/2],

we obtain that beginning with some T, |[beta]| > T,

|[K.sub.1/2 + i[beta]](x)| [less than or equal to] C(x)[e.sup.-[pi]|[[beta]|/2].

But this inequality is too rough and may be insufficient for conducting various proofs. To obtain more refined inequalities, we use [5]

(2.1) |[K.sub.i[beta]](x)| [less than or equal to] [Ax.sup.-1/4][e.sup.-[pi]|[beta]|/2],

where A is some positive constant, and the representations [8]

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

LEMMA 2.1. The following inequalities hold for x > 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [c.sub.0] and c are some positive constants.

Proof. We estimate the second additive term in (2.2), using the inequality (2.1),

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We next estimate the third additive term,

(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Combining the first term and estimates (2.3) and (2.4), we obtain the required inequality.

Furthermore, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For future use, an analysis of the behavior of the modified Bessel function [K.sub.[sigma] + i[beta]](x) for large values of [beta] is necessary.

LEMMA 2.2. For 0 [less than or equal to] [sigma] [less than or equal to] 1/2, |[beta]| [greater than or equal to] [[beta].sub.0] [greater than or equal to] 1, x [greater than or equal to] [x.sub.0] [greater than or equal to] 1, the following inequality holds,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [c.sub.1] > 0, [c.sub.2] > 0, [c.sub.1], [c.sub.2], [[beta].sub.0], [x.sub.0] are some constants.

Proof. We use the formula [6]

[K.sub.[micro]](x) = [pi]/2 sin([pi][micro]) ([I.sub.-[micro]](x) - [I.sub.[micro]] (x)), [micro], = [sigma] + i[beta].

1. We first estimate sin([pi][micro]). It is possible to show that for |[beta]| [greater than or equal to] [[beta].sup.(1).sub.0] > 0, [[beta].sup.(1).sub.0] some constant, the following inequality is valid

(2.5) [a.sub.1][e.sup.[pi]|[beta]|] [less than or equal to] |sin([pi][micro])| [less than or equal to]< [a.sub.2][e.sup.[pi]|[beta]|],

where [a.sub.1] > 0, [a.sub.2] > 0, [a.sub.1], [a.sub.2] are some constants.

2. We next estimate [I.sub.[micro]](x), [micro], = [sigma] + i[beta], [sigma] [greater than or equal to] 0. The following inequality is derived for [sigma] [greater than or equal to] 0 in [4]

|[I.sub.[micro]](x)| [less than or equal to] [I.sub.o](x) [(x/2).sup.[sigma]][e.sup.[pi]|[beta]|/2]/[GAMMA]([sigma] + 1).

It follows from the asymptotics of [I.sub.0](x) [10] for large x that for x [greater than or equal to] [x.sup.(1).sub.0] > 0, [x.sup.(1).sub.0] some constant,

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [a.sub.3] > 0, [a.sub.3] some constant.

3. We finally estimate [I.sub.-[micro]], (x), [micro], = [sigma] + i[beta], [sigma] [greater than or equal to] 0. Proceeding analogously [4], we can rewrite [I.sub.-[micro]](x) in the form

[I.sub.-[micro]](x) = [(x/2).sup.-[micro]]/[GAMMA](1- [micro]) [psi](x, [micro]),

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then |k - [micro]|= [square root of ([(k - [sigma]).sup.2] + [[beta].sup.2])] [greater than or equal to] k - 1 for 0 [less than or equal to] [sigma] [less than or equal to] 1/2, k = 2,3, ... , and [square root of ([(1 - [sigma]).sup.2] + [[beta].sup.2])] [greater than or equal to] 1/2, [beta] arbitrary. Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. We obtain, after some calculations, that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using for |[beta]| [greater than or equal to] [[beta].sup.(2).sub.0]) [greater than or equal to] 1, 0 [less than or equal to] [sigma] [less than or equal to] 1/2, the expansion of the gamma-function from [5] and the asymptotics [6] for [I.sub.1](x) we obtain that beginning with some [x.sup.(2).sub.0], x [greater than or equal to] [x.sup.(2).sub.0] [greater than or equal to] 1, the following estimation holds,

(2.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[a.sub.4] > 0, [a.sub.4] some constant.

Combining the estimations (2.5)-(2.7), we obtain that for 0 [less than or equal to] [sigma] [less than or equal to] 1/2, |[beta]| [greater than or equal to] [[beta].sub.0] [greater than or equal to] 1, x [greater than or equal to] [x.sub.0] [greater than or equal to] 1, [[beta].sub.0] = max([[beta].sup.(1).sub.0], [[beta].sup.(2).sub.0], [x.sub.0] = max([x.sup.(1).sub.0], [x.sup.(2).sub.0])) the following inequality is valid,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Denoting [c.sub.1] = [pi]/2[a.sub.1] [a.sub.3], [c.sub.2] = [pi]/2[a.sub.1] [a.sub.4], we obtain the statement of the lemma.

The properties of the modified Kontorovitch-Lebedev integral transforms are considered in [7]-[15].

3. Tau method approximation for modified Bessel function of imaginary order. Several approaches for the evaluation of the modified Bessel functions are elaborated in [1]-[2]. The Tau method [3] realization, with minimal residue choice for the determination of the polynomial approximations of the solutions of the second order differential equations with polynomial coefficients [16] of the following form

([a.sub.0][y.sup.2] + [a.sub.5]y)v"(y) + ([a.sub.1]y + [a.sub.2])v'(y) + [a.sub.3]v(y) = 0, v(0) = [a.sub.4], y [member of] [0,1],

is supposed. An n-th approximation of the solution is sought in the form of the n-th degree polynomial [v.sub.n](y), which is the solution of the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the coefficients [a.sub.i], i = 0, ... , 5, may be expressed by coefficients [b.sub.i], i = 0, ... , 5, [a.sub.n+2] = [sin.sup.2] [pi]/4(n+2)--the leftmost root of the shifted Chebyshev polynomial of the n + 2-th degree [T.sub.n+2]* (y) in the interval [0 ,1], [[tau].sub.n+2]--undefined coefficient.

The problem about determination of the polynomial [P.sub.n](y) = [n.summation over (k=0)] [p.sub.k.][y.sup.k], which is the least deviated from zero on the interval [0,1] among all n-th degree polynomials, satisfying the pair of linear correlations on the coefficients [p.sub.0] = 0, [n.summation over (i=1)][c.sup.(n).sub.i][p.sub.i] = 1 was considered. The following theorem is proved [16]:

THEOREM 3.1. If the sequence of numbers [c.sup.(n).sub.i], i = 1, ... , n, is alternating, then the polynomial [[tau].sub.n][T.sub.n]* [(1 - [[alpha].sub.n])y + [[alpha].sub.n]] is the polynomial least deviating from zero in the uniform metric on [0,1] among all polynomials of degree n, satisfying the indicated pair of linear relations.

On the basis of this theorem it's shown (as suggested by us) in the Tau method residue, in a number of significant cases, is a minimal in the uniform metric on [0,1], among all possible polynomial residues permitting the Volterra integral equations solution.

We have the following differential equation with polynomial coefficients for the approximation and computing of the second kind modified Bessel function [K.sub.i[beta]](x):

[y.sup.2]v"(y) + 2(y + 1)v'(y) + (1/4 + [[beta].sup.2])v(y) = 0,

v(0) = 1,

and the Volterra integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We obtain the following recurrence formulas for the coefficients of canonical polynomials [Q.sub.m](y) = [m.summation over (k=0)][q.sub.km][y.sup.k] in this case:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The minimality of the residue suggested by us follows from the Theorem 3.1 as qom/|qom| = [(-1).sup.m], m = 0,1, ...

The advantages of this modification, as compared with usual and other tau-methods, is shown.

4. Tau method approximation for modified Bessel function of complex order. A new numerical scheme of the Tau method application is proposed for the solution of the second order linear differential equations systems, with the second order polynomial coefficients of the following kind:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

in the unknown vector-function v(y) = ([v.sub.i] (y), ... , [v.sub.k](y)). It is assumed to have only one solution. Integrating twice and carrying an addition in the right part in the kind of the vector-polynomial [P.sub.n] (y), we derive for the determination of the n-th approximation of the solution v(y) = ([v.sub.1](y), ... , [v.sub.k] (y)) the system of Volterra integral equations with polynomial kernels

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the coefficients [b.sup.(j).sub.i] and [a.sup.(j).sub.i], i = 0, ... , 3k + 2 and j = 1, ... , k, are connected in a definite way and [P.sub.jn+2] (y), j = 1, ... , k, are n + 2-th degree polynomials. The different variables of the vector residue choice and its minimization are analyzed. The recurrence formulas for the canonical vector-polynomials coefficients convenient for the calculations are given.

Consider the system of two second order differential equations (k = 2) in more detail. This case is of particular interest for differential equations with complex coefficients. The scheme of the integral form of the Tau Method described in this paper can be used for deriving polynomial approximations of hypergeometric and confluent hypergeometric functions of the first kind with complex parameters.

The modified Kontorovich-Lebedev integral transforms [7] with kernels Re[K.sub.1/2 + i[beta]] (x) = [K.sub.1/2 + i[beta]] (x) + [K.sub.1/2 - i[beta]](x)/2 and Im[K.sub.1/2 + i[beta]](x) = [K.sub.1/2 + i[beta]](x) - [K.sub.1/2 - i[beta]],(x)/2i, where [ is MacDonald's function, is of great importance in solving some problems of mathematical physics, in particular mixed boundary value problems for the HELMHOLTz equation in wedge and cone domains. We find it necessary to compute Re[K.sub.1/2 + i[beta]](x) and Im[K.sub.1/2 + i[beta]](x) to use this transform in practice [13]. These functions also occur in solving some classes of dual integral equations with kernels which contain MacDonald's function of imaginary index [K.sub.i[beta]](x) [7]. Therefore, now we consider the second kind modified Bessel function [K.sub.1/2 + i[beta]](x) in more detail.

We have a system of two second order differential equations

[y.sup.2][v.sub.1]" + 2(y + 1)[v.sub.1]' + [[beta].sup.2][v.sub.1] + [beta][v.sub.2] = 0,

[y.sup.2][v.sub.2]" + 2(y + 1)[v.sub.2]' - [beta][v.sub.1] + [[beta].sup.2][v.sub.2] = 0,

[V.sub.1](0) = 1, [V.sub.2](0) = 0,

or the system of Volterra integral equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The following formulas for the coefficients of canonical vector-polynomials are derived [16]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

By means of computations is shown that the choice of the residue in the form [P.sub.jn+2](y) = [[tau].sub.jn+2][T.sub.n+2][(1 - [[alpha].sub.n+2])y + [[alpha]sub.n+2]], j = 1, 2, is optimal as compared with other known variants in this case too.

The applications for the numerical solution of boundary value problems in wedge domains are given in [18],[19].

Acknowledgments. The author wishes to thank the organizers of the Constructive Functions-Tech04 Conference for their encouragement and support.

Dedicated to Ed Saff on the occasion of his 60th birthday

* Received April 28, 2005. Accepted for publication December 22, 2005. Recommended by D. Lubinsky. The author's participation in the Constructive Functions-Tech04 Conference has been partially supported by the Russian Foundation of Basic Research, Grant 04-01-10790 and National Science Foundation, Grant DMS 0411729.

REFERENCES

[1] U. T. EHRENMARK, The numerical inversion of two classes of Kontorovich-Lebedev transforms by direct quadrature, J. Comput. Appl. Math., 61 (1995), pp. 43-72.

[2] B. R. FABIJONAS, D. L. LOZIER, AND J. M. RAPPOPORT, Algorithms and codes for the MacDonald function: Recent progress and comparisons, Journ. Comput. Appl. Math., 161 (2003), pp. 179-192.

[3] J. H. FREILICH AND E. L. ORTIZ, Numerical solution of systems of ordinary differential equations with the Tau Method, Math. Comp., 39 (1982), pp. 467-479.

[4] A. G. GRINBERG, The Selected Questions of the Mathematical Theory of the Electric and Magnetic Phenomenons, (in Russian), Publishing House of AN SSSR, Moscow-Leningrad, 1948.

[5] N. N. LEBEDEV, Some Integral Transforms of the Mathematical Physics, PhD Thesis, (in Russian), Ioffe Physics-Technical Institute, Leningrad, 1950.

[6] N. N. LEBEDEV, The Special Functions and Their Applications, 2nd edition, revised and complemented, (in Russian), Fizmatgiz, Moscow-Leningrad, 1963.

[7] N. N. LEBEDEV AND I. P. SKALSKAYA, The dual integral equations connected with the Kontorovitch-Lebedev transform, (in Russian), Prikl. Matem. and Meehan., 38 (1974), pp. 1090-1097.

[8] N. N. LEBEDEV AND I. P. SKALSKAYA, Some integral transforms related to Kontorovitch-Lebedev transforms, (in Russian), in The Questions of the Mathematical Physics, pp. 68-79, Nauka, Leningrad, 1976.

[9] A. F. NIKIFOROV AND V. B. UVAROV, The Special Functions of the Mathematical Physics, (in Russian), Nauka, Moscow, 1975.

[10] A. P. PRUDNIKOV, Y. A. BRYCHKOV, AND O.I. MARICHEV, The Integrals and Series, (in Russian), Nauka, Moscow, 1981.

[11] A. P. PRUDNIKOV, Y. A. BRYCHKOV, AND O.I. MARICHEV, The Integrals and Series, Special Functions, (in Russian), Nauka, Moscow, 1983.

[12] V. B. PORUCHIKOV AND J.M. RAPPOPORT, Inversion formulas for modified Kontorovitch-Lebedev transforms, (in Russian), Diff. Uravn., 20 (1984), pp. 542-546.

[13] J. M. RAPPOPORT, Tables of modified Bessel Junctions [K.sub.1/2 + i[beta]](x), (in Russian), Nauka, Moscow, 1979.

[14] J. M. RAPPOPORT, Some properties of modified Kontorovitch-Lebedev integral transforms, (in Russian), Diff. Uravn., 21 (1985), pp. 724-727.

[15] J. M. RAPPOPORT, Some results for modified Kontorovitch-Lebedev integral transforms, in Proceedings of the 7th International Colloquium on Finite or Infinite Dimensional Complex Analysis, Marcel Dekker, 2000, pp. 473-477.

[16] J. M. RAPPOPORT, The canonical vector-polynomials at computation of the Bessel functions of the complex order, Comput. Math. Appl., 41 (2001), pp. 399-406.

[17] J. M. RAPPOPORT, Some numerical quadrature algorithms for the computation of the MacDonald Junction, in Proceedings of the Third ISAAC Congress, Progress in Analysis, vol. 2, World Scientific Publishing, 2003, pp. 1223-1230.

[18] J. M. RAPPOPORT, Tau method and numerical solution of some mixed boundary value problems, in Proceedings of the Conference on Computational and Mathematical Methods on Science and Engineering CMMSE-2004, Angstrom Laboratory, 2004, pp. 236-242.

[19] J. M. RAPPOPORT, Dual integral equations for some mixed boundary value problems, in Advances in Analysis, Proceedings of the 4th ISAAC Congress, World Scientific Publishing, 2005, pp. 167-176,

[20] A. GIL, J. SEGURA AND N. M. TEMME, Evaluation of the modified Bessel function of the third kind of imaginary orders, J. Comput. Phys., 175 (2002), pp. 398-411.

[21] 1. N. SNEDDON, The Use of Integral Transforms, McGraw-Hill, New York, 1972.

[22] N. VIENER, Fourier Integral and Some Its Applications, (in Russian), Fizmatgiz, Moscow, 1963.

[23] M. 1. ZHURINA AND L. N. KARMAZINA, The Tables and Formulas for the Spherical Functions [P.sup.m.sub.-1/2 + i[tau]](z), (in Russian), VC AN SSSR, Moscow, 1962.

JURI M. RAPPOPORT Russian Academy of Sciences, Vlasov Street, Building 27, Apt. 8, Moscow 117335, Russia (jmrap@1andau.ac.ru).

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Author: | Rappoport, Juri M. |
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Publication: | Electronic Transactions on Numerical Analysis |

Date: | Dec 1, 2006 |

Words: | 4590 |

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