Expansões polinomiais das Funções de Bessel

7/25/2019 Expanses polinomiais das Funes de Bessel

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Polynomial Expansions of Bessel Functions andSome Associated Functions

By Jet Wimp

1. Introduction. In this paper we first determine representations for the Anger-Weber functions Jv(ax) and Ev(ax) in series of symmetric Jacobi polynomials.(These include Legendre and Chebyshev polynomials as special cases.) If v is aninteger, these become expansions for the Bessel function of the first kind, since]k(ax) = Jk(ax). In Section 3, corresponding representations are found for(ax)~"Jv(ax). Convenient error bounds are obtained for the above expansions. Inthe fourth section we determine the similar type expansions for the Bessel functions

Yk (ax) and Kk (ax). In Section 5, the coefficients of some of our expansions are tabu-lated for particularly important values of the various parameters.

2. Symmetric Jacobi Expansions of Anger-Weber Functions. A function f(x)satisfying certain conditions (for these consult [1]) may be expanded in the series

(2.1) /(*) = CnPnia-a)(x), -1 x 1, a > -1,

where Pn{a (x) is called the symmetric Jacobi polynomial of degree n. For ourpresent purposes we shall use a definition given in [2] :

(2.2) 2nn\Pn(a'a)(x) = (-)n(l - x2yaDn[(\ - x2)a+n).

Also

(2.3) C = h"1 /(*)(! - x2yPnla-"\x) dx,

(2.4) hn = ---y^--p- (u)M = -r^ , (v)o = 1.(n + + i)(n + a + 1)Q T(v)

Using the representation (2.2) in (2.3) and noticing that all derivatives of(1 x2)a+n up to and including the (n l)st vanish at x 1, we integrate

(2.3) n times by parts to get:

(2.5) Cn = (2nn\hnTl J fM(x)(l - x2)a+ndx.

Consider the integral definition of the Anger-Weber functions [2, v. 2, p. 35]

(2.6) Uax) + iEv(ax) = tT1 f eiiv^ttXBin*)o> = /(*).Jo

When v is an integer, J(ax) coincides with the Bessel function of the first kind

Max) [2, v. 2, p. 4].Now differentiate (2.6) n times under the integral sign, substitute the result in

Received February 27, 1962.

446

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use


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POLYNOMIAL EXPANSIONS 447

(2.5) and interchange the order of integration (which is, of course, permissible).The inner integral is known [3] and after evaluating it we have

C. = (-i)n(n + l)a(hn7rm)l/2\-l

(2.7)f eiH

Jo

$ j a sin ) d4>.

Use the power series expansion for the Bessel function in (2.7) and integrate term-by-term to get

(2.8) C = ( -i)n cos ^ + i sin ^ A Rn(v, , a

where

A =ann\

(2.9)|~|-l(?+i+Or(!-i + 1)

and R is conveniently described in hypergeometric notation [2, v. 1, p. 182] as

Rn(v,a,a) = jF8|| + -,|+ 1;

. 3 n , v . . n v , n a2"]

1.5 -5+ 1; - |-J

(2.10)

rv -I- M 4- 2'2 2 '2 2

Equating real and imaginary parts of (2.6) and (2.1), we get

J(oi) = ,AttPnla-aHx), -Uil(2.11)

(2.12)

where

(2.13)(2.14)

and

(2.15)

(2.16)

Ev(ax) = BP(a'a)0r), -lgigl,

4 = Aj?(u, a, a)v)= 0.

* The Bessel functions required to commute the coefficients in our expansions can be

systematically generated on electronic computers with the aid of techniques discussed in[6, 7, 8]. There are numerous tables available for hand calculations. The words "accuracy,""error," and "convergence" in this paper always refer to the properties of the expansionwhen truncated after a finite number of terms.



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may be truncated and rearranged in powers of x. Clenshtw [9], though, by using therecursion formulas satisfied by the Chebyshev polynomials, has formulated a con-

venient nesting procedure which allows one to utilize such expansions directly. Thescheme is as follows. Consider

(2.25) f\x) = AnmTn* () , 0 g x ^ a,n-o \a/

(2.26) f2)(x) = t An


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450 JET WIMP

and likewise for Bn . Also [2, v. 2, p. 206]

(2.36) max I Pnia'a)(x) I = (n + ") , ^ -\.

Letejv denote the error incurred by taking just N terms of (2.11) or (2.12). Becauseof the rapidity of convergence of the expansions, as shown by (2.35), the (N + l)thterm furnishes us with a convenient error estimate

a)i-| |JV Tya+(l/2) 1/2(237) MV-r(^ + 1)r(4^ + i)r( + i),I +where a * , N > v, 1 i 1.

Among the values of considered, it follows from (2.37) that the choice a = |,i.e., the Chebyshev case, yields the smallest error term for large N.

3. Expansions of Bessel Functions of the First Kind of Nonintegral Order.Results in the previous section gave symmetric Jacobi polynomial expansions forJ(ax) and Iv(ax) for integral v. When v is nonintegral, these functions are nolonger entire functions of x, and it is convenient to derive an expansion for the entirefunction

(3.1) T(u + l)(ax/2yju(ax) = 0Fi (v + 1; - ~Y

Corresponding expansions for T(v + 1) (ax/2) ~~"IV(ax) then follow, as before, from

(2.24).Let/(x) in (2.5) be the right-hand side of (3.1). Then we have

(3.2) Jv(ax) = (axY AnP&a)(x), -1 x S 1,n=0

where

A _(-)"(2a)2"_

n 2V2(2n + 2 + l)2(n + \)^,m/ 1 3 n2\

jjn+i; u + n+l, 2n + + t ; - a- \ .

These equations also follow from a result in [4]. Indeed, using a general expansiongiven there, an alternative formula for (3.3) can be stated. We have

(3.4) iF3

An =

2

PS


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For the Chebyshev case of (3.2) a = \ and

(3.6) Max) = (axY CnT2n(x), -1 g x rg 1,

where

. i'_^('^, /i^2 rn + |; v + n + 1, 2n + 1;(3.7) Cn = t"(~na/4),2"^1F2

2"n! r(u + n + 1)

Notice that when u = , (3.3) simplifies. Also, since

/ \-(l/2)

= I 1 cos (ax),(3.8) J-d/2)(aa;)

we infer the expansion

(3.9) cos (ax) = YJCnPana)(x), -liln=0

where

(_)V/22-"+(1/2,(2n + + |)(2n + + 1)(3.10) C = ,a+(l/2) J,2n+ u.

Concerning the optimum choice of a in (3.2), see the discussion surrounding (2.37).

4. Expansions of Bessel Functions of the Second Kind. The Bessel function andmodified Bessel function of the second kind are denoted by Yv(z) and Kv(z), re-

spectively, and a treatment of them can be found in [2, v. 2, Ch. VII]. If v is non-integral, then

(4.1) F(z) = [sin (wr)rV.(*) cos (wr) - /_(*)},

and

(4.2) K(z) = J [sin (wr)rtf-.() - /(a)]

so for such values of u expansions for the functions follow directly from the resultsof Section 3.

If v is an integer, it can be shown that

(4.3) Yk(ax) - | [


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452 JET WIMP

and

(

where

4.4) Kk(ax) = ( - )k+1 [y + In (^jj Ik(ax) - - i Nk_x(iax) + ^ Wh(iax),

(4.5)

and

(4.6)

Nk-i(ax) =

1 *1 / \2

V m-O \ /

(k -m - 1)!wi!

[0,

>0

fc= 0,

IF _\ v / v. (o.x\ m [hm+k + m]Wk(ax) = 2-, (-) Itt) ,/; i-\V

m=o \ 2 / m!(/c + ra)!In the above 7 = 0.57721 = Euler's constant and

(4.7) /im = l + i+---+i, o = 1m

We assume the value of log (ax/2) is known. Then, since expansions for Jk (ax)and Ik(ax) were found in Section 2, and since Nk-i(ax) is simply a polynomial inl/(aa;), we need expand only the entire part of (4.3), i.e., Wk(ax), in symmetricJacobi polynomials.

Using the representation (4.6) as/(a:) in formula (2.5), a straight-forward deriva-

tion gives the series

(4.8)

where

(4.9)

An

Wk(ax) = T,AnPn{a'a)(x), -1 S * 2 1,

[(-)* + (-)"](n++l)(n + + i)2"+2a+l

2 7">=0 /

V

(-)"(-*

m +

re n +

2m). /aV+2m [fc+m+ hm]m\(k + ra)!'

2 /n+0+1

We note that the expansion for Y0(ax) may also be obtained by partially differ-entiating (3.2) with respect to v since

(4.10) Yo(ax) = 2tt"-i)dJv(ax)

A similar procedure yields the expansion for K0(ax). The Jacobi series for Yk(ax)and Kk (ax) for k > 0, however, are not so easily obtained in this manner.

For re = 0 and 1, the Chebyshev cases of (4.3) and (4.4) are

(4.11) Yo(ax) = ? f7 + In f\ J0(ax) + En T2n(x), 0 < x ^ 1,

(4.12) Yax) bM")]j,(ax) - + Fn r2n+1(x), 0 < x 1,License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use


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(4.13) Ko(ax) = - y + hi fe) h(ax) + G T2n(x), 0


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N = [n(n + 2 + 1)] , y = (1 + 2a)/4. In general, if values of f(x) for com-plex x are desired, it is wisest to choose a such that the expansions are interpolatory

along a suitable ray in the complex x-plane and to stay as close as possible to thisray.

Suppose we have the truncated expansion

AT

(5.2) f(x) = ZAX(i) + e+1= 4>(x)+ i//+i, -1 * 1,n=0

and

oo

(5.3) ey+i = E Anrn(l)

Then


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458 JET WIMP

7. M. Goldstein, & R. M. Thaler, "Recurrence techniques for the calculation of Besselfunctions," MTAC, v. XIII, p. 102-108.

8. F. J. Corbat, & J. L. Uretsky, "Generation of spherical Bessel functions in digital

computers," J. Assoc Comp. Mach., VI, p. 366-375.9. C. W. Clenshaw, "A note on the summation of Chebyshev series," MTAC, v. IX, p.118-120.

10. Jerry L. Fields, & Yudell L. Luke, "Asymptotic expansions of a class of hyper-geometric polynomials with respect to the order I," to appear.

11. N. I. Achieser, Theory of Approximation, Ungar, New York, 1956, p. 57.


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Expansões polinomiais das Funções de Bessel