the zeros of hankel functions

Numerische Mathematik 7, 238--250 (1965)

The Zeros o f Hankel Functions as Functions o f Their Order

By JAMES ALAN COCHRAN

1. Introduction

The analysis of many physical problems requires an investigation of the zeros of specific transcendental functions. In situations having circular or spherical symmetry Bessel functions or combinations thereof are often involved. Recently various quantum mechanical applications have stimulated special interest in the roots of

(w)=0, (t.t) (d/aw) L (w) = o, (t.2)

(d/dw) H~ (w) + i WH x, (w) = 0 (W= constant), (t .3)

where these Hankel functions and their derivatives are to be considered as functions of their order v with fixed argument w.

In a t960 paper [1], MAGNUS and KOTIN investigated theoretically the behavior of the v-zeros of the Hankel function of the first kind, i.e. the roots of (t . t) . Somewhat later KELLER et al. [2] derived asymptotic expressions for the roots of (1.t), (1.2), and (1.3) when the fixed argument w was either small or large, as well as when [w[ was moderate in size but the desired root v had large modulus.

Many of the results of MAGNUS and KOTIN, as suggested by the authors themselves, can be extended beyond the restricted range of the argument w of H , 1 considered in their paper. Moreover, somewhat similar reasoning is ap- plicable in theoretical analyses of the v-zeros both of (d/dw) HI~ (w) and of linear combinations of this derivative and the Hankel function itself. We consider these matters in section 2 of this work.

The body of the paper is concerned with a systematic treatment of the roots of equations (1.1), (t.2), and (1.3) with large v and arbitrary w. Asymptotic expressions for these roots are derived in a unified manner from the expansions of OLVER [3], [4]. The results greatly improve upon the accuracy of the earlier expressions of KELLER ([2], p. 831 ) as well as, in a simp~fied form, eliminate insignificant discrepancies between the formulas of KELLER and those of MAGNUS ([I], p. 243).

For completeness, the argument principle is employed in a careful analysis of the asymptotic expansions derived herein. I t is proved that the expressions exhibited give the s m v-zero in order of magnitude as the integer index s tends to infinity.

The Zeros of Hankel Functions 239

We will have occasion to use the fact that the roots of equations (tA), (t.2) and (t.3) are symmetric about the origin, since e ~'=i H, 1 (w) and e ~=i(d/dw) 1 n~ (w) are even functions of v. Moreover, because of the relations

H~ (~) = H~ (w) (1.4) and

H ~ ( ~ ) = - - ~ . . . . ~ ( ~ ' w ) , (t.5)

where a bar denotes the complex conjugate of a quantity, results concerning the v-zeros of H i(w) may often be interpreted in terms of either H~ or of Hi itself with modified argument. These simple notions will be rather beneficial ill our investigations.

2. The e x i s t e n c e of v - z e r o s of (alaw) n~(w) Although H~ (w) is regular analytic only in the w-plane cut along say the

negative real axis from 0 to --0% it is an entire function of the complex variable v. Using HEINE'S formula (see [5], p. 21) for 0 < a r g w < ~ r , MAGNUS and KOTIN showed that the entire even function

g(v) - ] = e~(~+ll ~, Hi (w) o o

= f e,~ co,h, coshv t d t 0

is of order _~ 1. They then inferred that the Hankel function has a product expansion given by

(,- ;i ) where the vk are the v-zeros of Hi(w) for Im w>0 . Moreover, since g(v) is not a polynomial in v, there are a countably infinite number of zeros vk. MAGNUS and KOTIN easily derived similar results for positive real w through the use of NlCaOLSON'S expression (see [5], p. 54)

oo

"~'8 H~ (w) H~ (w) = f K 0 (2 w sinh t) cosh 2 v t dr, 0

where K o is the modified Hankel function of order zero.

The same technique can be applied to the function g(v) for " z c < a r g w < 0 when given by the companion formula due to HEINE ([53, p. 2t) :

CO

= f e - i " COSh t cosh (v t + v z~ i) d t + i f ei~ coS t cos v t d t . 0 0

Similar conclusions result, and the expression (2.t) is valid for [argw[ <z~.

The familiar recursion relation

2 (d]dw) H~ (w) 1 x =H~_x (w) - -H~+x(w)

240 JAMES ALAN COCHRAN:

permits the above considerations to be extended to the derivative of the Hankel function or, for that matter, to linear combinations of the derivative and the function itself. The order of the functions involved can be determined, and we conclude that

Theorem2.1. If [ a r g w ] < ~ , there are a countably infinite number of v-zeros of

a) (d/du,) H~(w) and of b) (d[dw) H~(w)+iWH~(w) Moreover,

(W = constant).

where the v~ are the v-zeros of (d/dw) Hi (w), with a similar result vatid for b).

3. The Asymptot ic Expans ions of the Hankel Funct ion

In numerous instances one is interested in the behavior of given functions for large values of their arguments or significant parameters. KELLER and his co-workers [2] made typical investigations of such properties in their asymptotic analyses of the roots of (t.t), (t.2) and (t.3). We would like to show that dis- tinctly improved results for the v-zeros with large v but moderate I w] as well as those with large argument w can be obtained in a systematic, unified manner from the asymptotic expansions of OLVER [3], [4].

The appropriate uniform representation for the Hankel function of the first kind takes the form ([4], p. 338)

H~ (v z) N 2e_~ni \ t --z~ ] 4 ~ ]t I A i (v ~#e~'i ~) ,=o ~' ~fi~) + A i' (v]#e ~"i ~) r~__~0 _~2(r~) } (3.t)

as Ivl->oo, 0_<argv=<z~, la g l< , where the original argument w of Hi has temporarily been replaced by v z and

with

and

r

B,(r = �89162 f t-i{](t) A,(t) -- A;'(t)} dt, O

Ao r = t , 1 t A,+I(~) = - -~e,(r + ~ f ](r B,(r dr

A , + I ( - - ~ ) = 0 for r > 0 ,

5 tz~(z~+4) /(~)---- 16r 4(t--z2) 8

(3.2)

(3.3)

1

f a, (3.4) $

The expression (3.1) is uniformly valid with respect to z in a region which in-

The Zeros of Hanke l Func t ions 241

cludes the sector [arg z[ ~z~-- ~ (6> 0). Moreover, when z is sufficiently near unity, (3.1) holds irrespective of the value of arg v.

A i(z) is the well-known Airy function of the first kind (see [6], pp. 94--97, for instance)

which satisfies the differential equation

d2W - - z W . d z ~

A i (z) is an entire function of the complex variable z.

The coefficients A,(~), B,($) defined by (3.2), /(~) from (3.3) and z(~) from (3.4) are all regular analytic functions of ~ over the whole ~-plane provided cuts are made along the rays arg~=-V�89 from ~=(3z~/2)}# : ~ i to infinity. D shall denote the open domain formed by the S-plane cut in this manner.

Now the v-zeros of/:I~ (v z) are given asymptotically in general by the v-zeros of the right hand side of (3A). These zeros in turn appear to be given by the v-solutions of

A i (v]e] ~i ~) = O,

from which we may infer that v~ :~'~Na, (3.5)

as I vl ~ oo where the as, s = 1, 2 . . . . . are the zeros of the Airy function A i. Owing to the nature of the relations (3.2) defining the coefficients A,(~) and B,(~), the results to be obtained later in the paper are completely compatible with this observation.

The zeros as of A i, fortunately, have been thoroughly studied [4, pp. 364--367], [7]. OLVER shows, for instance, that they are all negative real and may be expressed approximately by

(3.6)

for large index s.

If we now restore the original argument w = v z of the Hankel function, relation (3.5) shows that the v-zeros of Hi (w) should satisfy

w e_z~ i a s ~ ( v )

as Iv]-->oo where ~ and z are related by (3.4). Three limiting cases, therefore, could conceivably give rise to large values of v:

i) l e l ~ ~ t h I ~ g ( - r 1 6 7 and z(~) = ~(--r

ii) with I rgr z(O = 2e-~:-~ [l + e -~ : -~ + e -~ :o0 ) ] .


iii) 1r in D; then z(~) = t -- 2-/'$+i~o 2-~2+~aO(t).

Inspection shows that iii) is associated with the behavior of the v-zeros of H~ (w) for large ]w], and that ii) leads to a description of the large v-zeros for fixed more moderate [w]. We shall analyze these situations in turn.

4. The v-zeros of H i ( w ) for Large Iwl

Given w with [arg w[ < ~ and ] w [ > l , from (3.7) and iii) we obtain

W W

= w [ t + 2 - ~ + ~ ' o 0 ) ] (4.t)

= w + 2-~ a s e - ~ i w ~ + w - ~ 0 ( t ) *

as [ w I -+ 0% which clearly shows the linear dependence of the v-zeros for large [ w I.

If further accuracy is desired, additional information from the expansion (3.t) need be used. For instance, if

v, = w + 2 - ~ a, e- i '~iw ~ + w - ~ e + w-xO (t) (4.2)

as ]w[--> 0% with e to be determined, then ~ is given by

~ = w - ~ a , e - t = i + 2 ~ w - ~ [ e - - xg-a2-~(a,)~e -~=i] + w - ' O ( l ) . (4.3)

But equation (3A) implies that

A i (v~ e~ "~ r = v;- ~ 0 (t) which with (4.3) leads to

e = ~ 2 - t ( a s ) ~ e -~ ' i .

Higher order terms follow in the same consistent manner.

S. The Large v-zeros of H 1 (w) for Fixed w

The behavior of the large v-zeros when w is some fixed moderate value follows from (ii) and (3.7). We obtain

v , = z (r 2

where ~ satisfies the asymptotic equality

~',--, ~ (5.t) * KELLER and his co-workers have used a scaled version of the usual Airy func-

tion in their investigations. The identification qs=--3la, should be made when relating these results.


with cr given by the transcendental equation

0r ~ e~o:~ _ 2a~ e - h i = d ~ f . e w

(5.2)

The function cc can be approximately determined from (5.2) as

~ ~ _ 3 1 n [ ~ ] + lnlna, o[4~ 2 [ In d~ J in as -- '" '

(5.3)

as s--->oo so that the error term in (5.t) may be accurately specified. investigation of the original expansion (3.t) then shows that

�9 ln~ias O(1)

from which it follows that

1 V s = 2 W r 1

[ ln = as 1 - + - ~ U o 0)j

-- de-'' If-l- In'o% O(i)]

Further

(5.4)

as integer s-+ ~ . Since the v-zeros are symmetric about the origin v = O , there is another set of zeros given asymptotically by the negative of the right hand side of (5.4).

A simplified but less accurate expression for the large v-zeros of H~ (w) with fixed argument w may be obtained from (5.4) by first substituting the asymptotic expansion of (5.3) for ~ above, viz.

! d [ lnlnas O(t]] v s = - - e w ~ [ t + l n a s "-'J" 2 In ~r

Then using the representation of the Airy function zeros for large index (3.6), this last result can be recast as

__ iz~s [ i + ln lns O(t)]

tS- ~ W

as s--> co. The error term exhibited in the final expression (5.5) clearly shows that the discrepancies between earlier formulas of KELLER ([2], p. 83t) and MAGNUS ([1], p. 243), discrepancies which are at most of order t]ln s, are insignificant indeed.

I t is easy to conclude from (5.5) as did KELLER, that in general both Re % and Im v, become infinite as s--> oo. On the other hand, their relative order of growth is such that arg v, itself approaches ~ r as s increases without limit. Such values of argv, are compatible with the constraint 0_--<argv<~r expressed earlier for the applicability of the original asymptotic expansion (3.t). Moreover, a r g v ~ r leads to a bounded behavior for the coefficient functions A,(~), B,(~) as ]~]--> oo which establishes the consistency of our asymptotic analysis of these v-zeros.


6. The v-zeros of (d/dw) It~ (w) The asymptotic expansion (3.1) for the Hankel function may be differentiated

term by term. Analysis paralleling that of previous sections then shows that the ~-zeros of (d/dw) HX,(w) with large Iwt satisfy

v , = w + w) a,e-~ ~ t 30 5;; "~-w-IO(]) (6.t)

as [w[-+ oo, where the a: are the stationary values of the Airy function A i. The analogue of (5.4) for the large v-zeros of the derivative with fixed moderate

ze is [ ln~a~ ] , I well,+1 1 + - - 0 ( 1 ) (6.2) v, ----- ~ (a;p

as integer s ~ oo, where

/3~ e ~ _ 2(a;)~ e-=~ (6.3) e w

In a manner identical with that of the preceding section 5, the expansion (6.2) likewise can be expressed in a simplified but less accurate form using the asymptotic representation of/3 in terms of s. Since ([4], p. 367)

]' g = - s [~ +s - lO0) ]

= a , O + s - l O 0 ) ]

as s---> ~ , the result obtained by this procedure differs from (5.5) by less than the error term there exhibited.

7. Linear Combinations of Hankel Function and Derivative

The investigation of the large v-zeros of the linear combination

(d]dw) H~(w) + i WH~ (w) (W = constant) (7.1)

is in complete harmony with the analyses of previous sections. Careful consider- ation of the significant terms in the asymptotic expression which results from substitution of the expansion (3.t) in the above combination shows firstly that when w remains bounded, the v-zeros v~ of (7.1) are asymptotically the same as those of the derivative (d]dw) Hi (w) itself. In fact the zeros satisfy

~ = v ; [ t + ln,~; (~-;)3 0 (t)] (7.2)

as s--> 0o, with the ~,' given by equation (6.2) of the preceding section. On the other hand, for large ]w[ the g are more closely approximated by the

v-solutions of

e~'~ d-~ A i'(# e t ~ ) + i W# A i (# e~ " ~ ) = 0. (7.3)

The Zeros of Hankel Functions

Characteristically simple analysis shows, indeed, tha t the zeros satisfy

where c, is the s th root of the equation

A i'(c,) = (-~w)i W e - ~ " A i(cs).

As ]w I -v 0% the values c s tend towards the Airy function zeros as, viz.

2 ~e _~ c~=a~ + w 0 ) .

2 4 5

(7.4)

8. The Accuracy of Indexing the ~-zeros

I t is rather natural for one to inquire whether the asymptotic expansions (5.4), (6.2), and (7.2) which have been derived for the large roots of equations (t . t) , (1.2), and (t.3) respectively, with fixed w, actually give the s th root in order of magnitude as the integer index s tends to infinity. Since it is not a trivial mat ter to verify this assumption, we shall take the time to carefully establish its validity for the v-zeros of H~(w). Obvious modifications can be made in the analysis presented within this section in order to prove the corresponding result for the v-zeros of either the derivative alone or linear combinations of the derivative and the Hankel function itself.

Our proof is organized along the following lines. We consider the contour in the v-plane given by

where fl satisfies equation (6.3), namely

3~ e ~ _ 2(a~)~ e - "~ (6.3) e w

As the integer index s becomes increasingly large, it is verified that

where v, is given by equation (5.4), so tha t in the limit the curve ff separates the v-zeros with index s - - i from those with index s. We also establish that for given w and large enough s, H~ (w) does not vanish on c~. Since the contour c~ is independent of the argument of w, this allows us to infer that given the modulus of w, the number of v-zeros of H~ (w) within ~ likewise is independent of arg w. We then inspect the special case of arg w=~z~, enumerate the zeros within the curve cg and, on the basis of this result, conclude that our asymptotic expansions actually give the s th v-zeros in order of magnitude as the index s tends to infinity.

Details of the argument just sketched are given in a series of lemmas followed by the proof of the main theorem of this section.

Numer. Math. Bd. 7 ] 7

246 JAMES ALAN COCHRAN."

Lemma 8.1. If v, and/5 are given by equations (5.4) and (6.3) respectively, then

for all sufficiently large index s.

Proo/. Let fl~=O #% Equation (6.3) can then be written as

0 e~%xP [w ei~] = R e i~ which leads to

e e ~ ~

w ~ sin ~ = p.v. (~9-- 9)"

Equation (5.2) defining ~ is of the same form as (5.4) and can be expressed in a similar manner. From these relations we may easily verify that do]dR and (d/dR) [R/Q] are both positive as s (and hence R) tends to infinity. In fact we have

i)

and

R = ~ [t -]-s-lO(t)],

ii) dR .= e~w[ [t + s-lO (t)] ds

. , / s

iv) do __ e[w[ [t + ln-~O(t)] dR 5 - ~

v) a a (1)] a s S .--~ ~ o

Since I,,-11 < I<1 < I,,I for all s (see [7]), it follows that

I exp(w < I exp(w < I exp(w (8.2)

This relation then implies

Iv,-ll < 1�89 < l,',l (s--, =) (8.3)

by virtue of the fact that from i)--v), the difference between corresponding terms in (8.2) and (8.3) is less than the difference between successive terms in either expression for sufficiently large index s. Q.E.D.

If a circular contour $' in the v-plane is given by the relation (8.t), then the essence of the above result is that in the limit of large s the curve c~ separates the v-zeros of H~ (w) with index s - - t from those with index s. In fact, it may be shown that

I ,emma 8.2. Given w with larg w[ <~r, for fixed large enough s the Hankel function H, t (w) is bounded away from zero on ~.


Pro@ Let z(~)= w and parametrize the circular contour cg with the real variable co by

v = �89 e w e~ ~ ei% (8.t') If we define Z by

- - Z = # e l " i ~ ,

then from section 6, we have

w = w e iO( - a;)~ + v o~ + v- lo (1).

The asymptotic expansion (3.t) is valid for 0=< arg v=< z, which implies that co should satisfy

9 - � 8 9 1 8 9 (8.4)

where 9 = arg/5t. Since f = (In s)-lO (t) as s-+ oo from the previous lemma, the appropriate asymptotic expressions for the Airy functions explicitly appearing in (3.t) are (see [4], p. 364)

i) A i ( - - Z ) = ~ {cos ( 3 Z ~ - - 4 @ [t + Z-30 (t)] + Z -~0 (t)sin ( 3 Z 8 - - 4 ~ ) }

and Z i

ii) A i '(-- Z ) = ~ - { s i n ( 3 Z t - - 4 ~) [t + Z-80 (1)] + Z-~O (1)cos ( 3 Z ~ - 4:~)}

Now c o s ( 3 Z , _ 4 ~ ) = c o s A [ t + ~ O ( t ) l + s i n A lnSs 0 ( t ) ,

(8.5) in 2 s In

sin ( 3 Zt -- �88 zc) -= sin A [ t + ~ z - O ( l ) ] + c o s A s s 0(t)

as s--> 0% with A = ~ ( s - - ~) e ~ ' - -~:~+vo~,

= ~s [ t + l n l n s 0 ( l ) ] I t is straight- where we have used that fact that Iv[ In s [ ~ "

forward to verify that, in the range given by (8.4), cos A is bounded away from zero and [tan A[ is bounded from above for large enough s.

In view of the above expressions we may conclude that

and

Z - t [ lns 0(t)] i') A i ( - - Z ) = ~ - c o s A t + ~

Z ~ ii') Ai'(--Z) = ~.~ cosA 0(1)

i~) A i ' ( --z) = Z~O0) a i ( - z )

= s tOO).

If we substitute these results in (3.t), recognizing the bounded nature of the t7"


coefficients A,(r B,(r we finally obtain

e-�88 [ t " ]n~s 0(t)] HI~(W)-- v �89 cosA + - ~ -

as s--> oo. Owing to the behavior of the cosine term, this last relation shows that for fixed large enough s, HI~ (w) is bounded away from zero on that half of ~ satisfying (8.4). The same conclusion follows for the remainder of c~ by symmetry. Q.E.D.

Once the nonvanishing character of Hlv (W) on q~ has been established, it is then a simple consequence of the argument principle that

Lemma 8.3. The number of v-zeros of H~(w) within cg is independent of w.

Proo/. Let N(w) designate the number of v-zeros of H~ (w) within the curve c~ of (8.1) denoted by Iv[ =r(w) . Forasmuch as Hi(w) is an entire function of v and regular analytic in w, the differential quotient

AN __ t [ N ( w + A w ) - - N ( w ) ] Aw Aw

may be expressed as

AN t I f d[logH~(w)]-- f d[logH~(w)]} + A w -- Aw- ~l~[='(w+~w) [vl=r(w)

+ f d[ /H (w)]+Remainder.

Since r is a continuous function of w, we can infer from Lemma 8.2 that for all small enough Aw, Hi(w) does not vanish on the contour ]v] =r(w+Aw). The remainder term can thus be estimated as AwO(a). Moreover, if 1 H~(w) :#0 on Iv I =r(w+Aw), then dH*~(W)/S~(w)-- is a continuous function of v on the dw I

entire contour, and the value of the third integral above is identically zero. The first term likewise contributes nothing in the limit A w-> 0 since the number of v-zeros of 1 H~ (w) within the circles ] v[ = r (w + A w) and [ v [ = r (w) are equal or all small enough A w. Hence,

AN=AwO(t) as Aw-+O, Aw and

d NN ~_ 0. Q.E.D. dw

The contour cg given by the relation (8.t) is actually independent of arg w. Thus, the modulus of w completely determines, for given s, the position of the curve ~ in the v-plane. The above lemma (a special case of a far more general result) then implies that, given I wl, the number of v-zeros of H~ (w) within the now fixed contour (g is independent of arg w. We may enumerate these zeros, therefore, by analysis of Hi(w) for any value of arg w with [arg w[ < z~. We choose the special case arg w = ]z~ and employ the argument principle to show that

T h e Zeros of H a n k e l F u n c t i o n s 249

Theorem 8.1. There are precisely 2 (s -- t) v-zeros of HI~ (w) within the contour ~' of (8.1) for all sufficiently large index s.

Pro@ From the proof of Lemma 8.2 we know that along the contour

- - v~g5

is valid, as s-+oo, for - - ~ o ~ since a r g w = ~ implies 9=arg/~----0. Use of the argument principle necessitates the investigation of the phase variation of each of the terms of this expression (8.6).

The most cumbersome term has the form

arg cosA =- - tan-1 {tan R tanh I} with

I = ~ ( s - ~) sino~ + Ivl o) cos o .

As co varies from -- ~ to + ~ we obtain twice the angular variation experienced in the range 0=<o=<~. Over this latter interval we have I=>0 varying from 0 to ~(s--~) and R varying monotonically from ~( s - - t ) to - - ~ (iv I +1) . The phase variation of this term can be calculated, therefore, as

A {arg cosA} -= 2 [~ (s -- 1) + ~ ~ (I vl + ~)~ + e-2'O (t).

Returning to the entire expression (8.6) we obtain

In ~ s . , A {arg Ee'~'~iH 1, (w)]} ~- -- -~-~ I vl ~ [sin col -- !zJ2 co + A {arg cos A} -t- - - ~ - 0 (1)

= 2 e ( s - - t ) ' ln~s o + - 7 - (1) ( s ~ oo)

for --~-~--<o<�89 In view of the evenness of e~*~iH~(w) as a function of v and the nonvanishing character of the exponential factor, the above result leads to the number of v-zeros of H~ (w) within ~ as

1 d{argH~(w)} =

4

= 2 ( s - t )

for all sufficiently large index s. Q.E.D. Recall that the v-zeros of H I (w) are symmetric about the origin. Since,

as we have noted previously, the contour ~ separates the v-zeros of H~ (w) with index s - - t from those with index s, we finally may conclude that

250 JAMES ALAN COCHRAN: The Zeros of Hankel Functions

Coro l la ry . The a sympto t i c expansion (5.4) ac tua l ly gives the S th v-zeros of H~ (w), in o rder of magni tude , as the index s tends to inf ini ty.

References [1] MAGNUS, W., and L. KOTIN: The Zeros of the Hankel Function as a Funct ion

of its Order. Numerische Math. 2, 228--244 (t960). [2] KELLER, J. , S. I. RUBINOW, a n d M. GOLDSTEIN: Zeros of H a n k e l F u n c t i o n s a n d

Poles of Scattering Amplitudes. J, of Mathematical Physics 4, 829--832 (1963). [3] OLVER, F. W. J . : The Asymptot ic Expansion of Linear Differential Equations

of the Second Order for Large Values of a Parameter, Phil. Trans. A 247, 307--327 (1954).

[4] - - The Asymptot ic Expansion of Bessel Functions of Large Order. Phil. Trans. A 247, 328--368 0954).

[5] ERDELYI, A., Edi tor : Higher Transcendental Functions II , Bateman Manuscript Project. New York: McGraw Hill Book Co. 1953.

[6] - - Asymptot ic Expansions. Dover Publications, Inc., New York 1956. [7] MILLER, J. C. P.: The Airy Integral. British Association for the Advancement

of Science, Mathematical Tables, Part-Vol. B. Cambridge 1946.

Bell Telephone Laboratories, Inc. Whippany, New Jersey 07 98t (USA)

(Received October 28, 1964)