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Numerische Mathematik 2, 228--244 (1960)
The Zeros o f the Hankel Funct ion as a Funct ion o f its
Order*
By
WILHELM MAGNUS and LEON KOTIN
1. Introduction
The theory of diffraction of electromagnetic waves by a sphere
requires a knowledge of the zeros of certain transcendental
functions.
In the simplest case of a perfectly conducting sphere of radius
a, the equation in question is H l(ka) = 0, where H 1 is the Hankel
function of the first kind of order v, and k is the wave number of
the incident wave. The quant i ty to be determined is the order v,
with ka being regarded as a parameter. Special attent ion is paid
to the case in which z=ka is real.
We evaluate v only for sufficiently large values. For other
values, we in- vest igate the behavior of v as a function of the
argument. Specifically, we derive certain inequalit ies satisf ied
by the real and imaginary parts of v and the argu- ment z; we
investigate the behavior of v as z--~0; and we obtain an
expression
for g~. dz
Although many of the results can be extended, the argument z of
H i will
be restr icted for the most part to the right half plane; i.e.,
]argz]
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The Zeros of the Hankel Function 229
order of an entire function to prove that there exists an
infinite number of zeros, and also to obtain an infinite product
representation.
From HEINE'S formula (see [1], p. 26), for 0< arg z<
z~,
oo
g(v) ~ ~2 e("+l)~ii2 Hl'(z) = f e~:c~176 0
I f we let Ivl =~, oo
I g (~')1
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230 WILHELM MAGNUS and LEON I~OTIN:
We can extend this result to the real axis by applying the same
technique to the function h(v) defined below. When x>0, from
NICHOLSON'S formula (U], P. 31),
(2.~) h(,) = v =' ~ (~) m (~) = f ~o(2z sinh t) cosh 2v t dt
o
where K o is the modified Hankel function of order zero. Then if
we put Iv] = ~, OO
lh(v)[ < f [Ko(2zsinht)I e2O' dt. o
Now for any v and z for which x>0 ([3], p. 181), OO
(2.2) K~ (z) = f e -~ ~o~h, cosh v s ds. 0
Therefore,
Thus
O0 CO
IKo(2z sinh t)] ~ f e--2*~h~~ ds
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The Zeros of the Hankel Function 231
for x>O, where the vk are now the zeros of 1 H.H,. Moreover,
equating coeffi- cients of v 2, we have
oo
2f t 2 K 0 (2z sinh t) dt I _ h" (o ) _ 0
oo
Z v~ 2h(0) f Ko(2zsinht ) dt 0
For z real and positive, if v (i) is a zero of H{, we have @)=v~
2). Hence
and
Theorem 2.2.
k
oo
f t2Ko(2xsinht) dt o
oo
f Ko(2xsinht ) dt 0
To show that H~ has only simple zeros, we require the following
well-known
Lemma 2.1. Let F(t) be positive, continuous, and decreasing for
t>0. Then oo
f F(t) sin t dt > O. 0
Pro@
f F(t) sin t dt = sin t dt + f F(t) sin t dt , 0 n=0L 2n~
(2n+l)r~
By the first mean value theorem,
(2 n+l ) ~ (2 n+ 2)
f F(t) s intdt+ f F(t)sintdt-~2F(t')--2F(t"), 2n~ (2n+l )~
for some t', t" such that 2n;z=O, p(t )>0, p ' ( t )o,
t-~q-l(x)=t(x). Then OO OO
fp( t ) sinq(t ) dt = /" pit(x)] sin xdx. J q'[t(x)]
0 0
Applying Lemma 2.t to the function F(x)= P- completes the proof.
q' Now we can prove the simplicity of the zeros. For if we let H~
(z)= 0 where
x>0, from a formula of WATSON (see [1], p. 30)
(20
H~(z) oI4~(z)ov 8j~ f Ko(2zsinht)e_~ dt. 0
17"
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232 WILHELM MAGNUS add LEON KOTIN:
Since H~ (z) =~0, using (2.2) it suffices to show that
0 (see Theorems 3.3 and 4A), the integrand, as a function of t,
satisfies the requirements
of Lemma 2.2 and is therefore positive for each s. Consequently
OHm(z) ~0 Ov and we have
Theorem 2.3. The zeros of H}(z) are simple when x>0 and
y>--0. We note that the proof is valid for the general
cylindrical function of the
form aJ~ (z)+ b Y~ (z) in which the coefficients are independent
of v.
3. Zeros of H~(z) for a fixed, complex
In this section we shall derive several inequalities relating
the zeros v and the complex arguments z. For the most part, z will
be restricted to the upper half plane. If we include the positive
real axis this corresponds to the actual physical situation, since
the wave number, which is essentially z, lies in the first
quadrant.
Theorem 3.1. Given f l~0 , there exists a positive y such that
H~(iy)~O. Proo/. For b real, we have (see Eli, p. 34)
oo
f e-t cosh b dt 2i e ~ ~I2 sin (fl b) HIS (i t) sinh fl ~sinh b
"
0
Then the integral vanishes. Since Choose/3 = b"
it is pure imaginary, and must vanish for some t> O.
In this case we can show that ]fi] >y by means of the
following classical argument. The function H~ (ze t) =u (t) +iv (t)
satisfies the equation
a2 H~ (~ ~') + (~ ~' -- ~) H~ (~ ~') = 0 dl2
If we multiply by H~ (ze% separate the real and imaginary parts,
and integrate, we obtain for y>O
oo
(3.1) f (~ y ~ ' - ~t~) (~+ ~) dt =o to
where t o is a root of u(t)+iv(t) and OO OO
(3.2) f[(x2--y2) e~t--(o~--fln)l(u2+v2)dt=f(u'2+v'e)dt>O. to
to
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The Zeros of the Hankel Function 233
We now let z=v. Then equation (3.t) becomes 0o
/3f _ t) (u2 + dt = 0. to
If to_>0, then we must have ef t=0. But then ~=0 s ince
/3=y>0, and (3.2) becomes
oo
f (1 - (,3 + d, > o. to
This, however, is impossible if t0>0. Therefore, t o must be
negative. We have thus obtained the following result.
Theorem 3.2. If H ) (av)=0 for a>0 and /3>0, then ao, then
I/3I >y . We can exploit equation (3.t) somewhat further. For
instance let x>0,
and recall that y>0. If t is sufficiently large,
xye2t--o~/3>O. Then for the integral in (3.1) to vanish, since
the above function is monotonic in t, we must have xye~to--~/30 are
pure imaginary. If we now consider equation (3.2) and assume x 2 --
y=< 0, the previous reason-
ing indicates that (x 2- y2)e2t.--((X2--/32)>0. This is the
content of Theorem 3.4. If H}(z)=0 with x2O, then o:2--/32
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234 WILHELM MAGNUS and LEON KOTIN:
The derivation of this formula for real values of v is
reproduced in WATSON (E3~, pp. 21t--212). This can be extended to
complex values by only slight changes. Without going through this
lengthy analysis, we merely state the following inequality for R.
If 21zl=2r>=~-- 89 , fl~O, and x>0, then
= 2r - t 2 r ] / ' (v+ 89 "
But it can easily be shown that when (~+ t) (~ -- 89 89
fl=>0, and x>0, then
2lz [
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The Zeros of the Hankel Function 2}5
or eo
x
from the behavior of the Hankel functions at infinity, and the
fact that
H~ (0 =H~ (0. Since the last integral converges, we have
Theorem 4.1. If H~(x)=0 for x>0, then ~f l>0. As far as ~
is concerned, this tells us only that l~l> 0. We can improve
this
inequality, but first we require the following well-known
extension of Lemma 2.]. co
Lemma 4.2. Let f / ( t ) cos bt dt exist for b~0, where/(t ) is
twice differenri- 0 co
able for t>0 with ['(t)
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236 WILHELM MAGNUS and LEON KOTIN:
Then
(4A)
oo
cos g 7~ K~:(2xs inht) cos2f l td t=~ 0
oo
f ~' (4.2) sinc~z K~(2xs inht )s in2 f l td t=~6
(IH~l~e~=--IHil2e-~" ) . 0
After subtracting we obtain
co
=' f K2~ (2 x sinh t) cos (2fl t + ~ 7c) dr. 8 IHi(x)l~e-a~ =
0
After dividing both sides by cos ~, integrating by parts, and
applying equation (2.2) for K2~ we find that when x>f l tan~+~,
Lemma2A is applicable.
We can close the ~ interval under consideration in the following
manner. If we let v = 89 + ifl, then
4 9 1 2 - H=_~ H~) = i(Hi H~ -- H*~ H~) = IHil ~ e-as + IH~I ~ ~
- - * (H~ H~- I 1 2 ~t
where the argument of both Hankel functions is t. This is as
easily proved for ~----- 89 and incidentally provides an extension
of (4.1). Now if we divide both sides by t and integrate from x to
o% applying Lemma 4.1 we find
co
x
whence
1 Then ]f l )> x . Theorem 4.4. Let Hi (x )=0 with x>O, a=
~: ~ . 2
If we apply Lemma 2A to equation (4.2), we obtain
Theorem 4.5. I f x>0 and ]=[< 89 then
sgn [[H~ (x) I e ~= -- ]H 1 (x)]] = sgn ~ ft.
5. An application of HURWITZ'S Theorem
We can extend the resu]ts of the preceding section by using the
following theorem due to HURWlTZ (I2], p. tt9).
Theorem 5.1. Let / , (z ) be a ~equence of analytic functions
which converge uniformly to /(z) ~ 0. Then z o is a zero of /(z) if
and only if it is a limit point of the set of zeros of ]n(z).
In particular, if ],(z) is the partial sum of a power series, we
can use the following result due to CAUCHY to get a bound on the
zeros of the limit.
Lemma 5.1. Let /~(z )~ao+alz+a2z2+. . .+a ,z"=O with ao~0.
Then
[ ~ iI -~ I~ l~ ~+~maxl~; 9
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The Zeros of the Hankel Function 23 7
Now for x > 0,
where
oo
-g-=~ H ~, ( x) tt~ ( x) = f Ko ( 2 x sinh t) cosh 2 v t d t
0
~, an~ 2n n=O
oo
' f< ~ a, (2 n) ! 2 t) (2 x sinh l) d t o
after reversing the order of summation and integration. Applying
HURWITZ'S Theorem, we obtain
Lemma 5.2. If H~(x)=0 and x>0, then
and
We note that
1 max a,,] -~ . [~1 > t+ ~o
oo
a,~ < f K o (2 x sinh t) cosh 2 t dt 0
__ 2~ 2
s [HI(*)I~
yg$
ao = ~ l~0 ~ (x)l ~
This results in the following simpler inequality.
Theorem 5.2. If H~(x)=0 and x>O, then
[ tti (x) 3]-~ i~l >t l+ Ha*i- ] '
which implies
I~1 > 1+ ~ and, if x > {,
[ i~ ' l>t t+ s . -~ ] "
The latter inequalities are obtained from an application of the
asymptotic formula (3.3) and (3.4).
These inequalities can be improved at the expense of the range
of x.
Theorem 5.3. Let H~(x)=0. If x=>0.6312 then
2 --,}
xiHo /x>l ] .Proo]. I t can be shown that arc sinh s=0.
Then
oo
~"" f s~""Ko(2Xs)ds. 0
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238 WILHELM MAGNUS and LEON KOTIN:
According to [3], p. 388, this is
__ 2 2n-2 /,2 (k n + fl_) = bn" (2n) l x 2k~+l
Forming b~+l/b n we find from the fact that F(x) increases for
x~t .5 that b,+:0.5. For n=0, b:>b o if xe~=>0.606 . . . .
2F~(k+{)jm Thus a~O, co
igl(x)[~ = s f Ko(2xsinht ) cosh 2tdt 0
co
> -:~ fK~ c~ 0
2 - - :7"~X
6. Behav ior of the zeros as x -~ 0
The inequalities [ e [ > x -V~//2 and x
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The Zeros of the Hankel Function 239
Since~--~0, IF(v)l ,~ [F(-v)I and we find that lim ~log ~- + =0.
It remains
to show that fl-+0, for then F ( f )~ 1 and the proof would be
complete. This is shown in *'
Theorem 6.2. Let H~(x)=0 with x---~0+. Then v--~0. Moreover for
some integer k, fl log x + k ~ = o (v x).
Proo/. We know that ~--~0. Suppose fl-r Then since fl decreases
with x (Corollary 7A), fl approaches a limit flo which we may
assume positive. Applying
(~)~ ,-~e -~12, from the proof of the preceding theorem, to
(6.2), we obtain
ea'~/~I~(ifl~ [c~ flol~ 2) - is in( fl~176 2)] +
+e~~176 [c~ flol~ 2) + i sin (fl~ l~ =o(x) .
Choose a sequence of x's converging to zero such that sin (rio
log -2-) = 0 for each x. k
Then ~(iflo ) +F( - - i f l o ) =0. Similarly choose another
sequence of x's such that
c~176176 =0. Then F( i f lo ) - -F ( - - i f l o )=O. These
results are incompatible; t
thus fl0=O.
(2) Therefore from Theorem6.1 we have Also I"(v) N v . from
(6.2)
= :
= -- ~2: sin (fl log 2)
$'
for some integer k, depending on which zero v we take. Then/~
log x + k 7r--~ o (vk).
7. The derivative of v In this section, we shall derive a
formula which was first obtained by ~VATSON
(see [3J, p. 508), oo dc= f dv 2c Ko(2cs inht ) e-2~tdt ,
0
where v is real and c is a positive zero (in z) of the
cylindrical function ~(z) coss+Y~(z)sins for constant s. This form
immediately excludes H i. We shall generalize it to apply to
complex values of z and v, as well as to any function of the form
C, (z) = clJ~ (z) + c 2 Y, (z), where q and c 2 are independent of
both z and r.
All the results of this section will be corollaries of Theorem
7.1. If C, (z) is of the above form, with x> 0, then any zero v
satisfies
dv [ 'K (2zsinht) e-2~tdt = t (7A) 2z ~. ] o 9 0
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240 WILHELM MAGNUS and LEON KOTIN:
Proo/. Let B, (z) = bl J~ (z) + b 2 Y~ (z) be a function of the
same type as C, (z) and linearly independent of it. Then
6 = b 1 c 2 - - b2 c l =4= 0 .
The Wronskians of these functions with respect to v and z are
respectively
B ec , C~ aB, _ ~ { , ~Y,, aj~ " av - - av \J" Tv-- - - Y" av )
oo
4af - - 7~ K 0 (2zsinh t) e-2~r 0
([1], p. 30), and
If C, (z) = 0, then
dC, __ C~ dBv 26 B~ dz dz -- :~z
(7.2)
and
(7.3)
B dC,. dv aC~ v dz +B~ - -0 , dz ~v
dd~ 4=0. Each term can now be replaced by one of the above
Wronskians, whence
giving us the theorem.
We shall restrict C~ to H~, although many of the following
results are valid for the general cylindrical function.
Corol lary 7.1. Let H~ (x )= 0 for a and x positive. Then
~'(x)> 0 when a~ or x>=~; and f l '(x)>O.
t~roo/. Solving (7.t) for dd~ and separating real and imaginary
parts, we find
oo
dOt- = 22 -dv i2 f Ko(2xs inht ) e -2~'cos2t3tdt dx ax I d
0
oo
d3 dv 2 ~.. =2x ~ f Ko(2xsinht ) e-2~s in2f l td t . 0
I f we express the kernel as
/(t) =- K0(2x sinh t ) e -2~t oo
(7.4) = fexp I - - 2x sinh t cosh s -- 2~ tJ ds, 0
the corollary thus follows from an application of Lemmas 4.2 and
2.1 respectively.
Moreover, with ~>0, we have from (7.t) and (2.1)
oo
t
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The Zeros of the Hankel Function 24t
Corollary 7.2. If H i (x) = 0 with x>O, then
which implies [/(x)[ > 2
We note from the first inequality that l im[/ (x) l =o0.
X-+0
A similar inequality is obtained from (7.2) and (7.3). For from
these,
[~'l = 2xlv'12 f Ko(2xt) e -~t dt 0
~-- -1 arccosh~x if e>x
o;~ ~-89 = lv ' l 2 t - x2} arc cOS-x, if ~ x.
Corol lary 7.3. If Hf f (x)=0 with x>0, then
dv _> sin(h),2y
If we consider the k-th zero ~, and let k--~ 0% since ~k-+ o0
(see section 8) we find that lim [v~[ = o~.
k--+ oo
These conditions on [v'l give us no picture of the trace of the
zeros in the v-plane as x increases. We can remedy this by
considering/5 as a function of
a~ and determining the behavior of d~ "
Corol lary 7.4. Let H~(x)=0 with x>0 and ~>{. Then
a~ < 2~
-as = Vso;- I
Proo/. From (7.2) to (7.4) ~'/(t) si~ 2~tdt dfl =
do: oo y /(t) cos 2fl t dt 0
dfl where ~>{. After integrating by parts, we find that do; V
~- 1"
We note that this value exceeds the preceding one, and applying
Lemma 2A we d~ find that d~ < 2/3 ~. This implies the
corollary.
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242 WILHELM MAaNVS and LEON I{OTIN:
These results have so far been restricted to real values x. For
complex values z, we get the following analogue of Corollary
7.t.
Corollary 7.5. Let H~(z)=0 with x, y, 0~, fl positive. Then
x~x -- y~x >0 when c~>~;
and am x~+y~>0.
Proo/. From (2.2) for x>0 oo oooo ooo0
f K 0 (2 z sinh t) e -~t dt = f f exp [2 z sinh t cosh s -- 2 v
t_] ds dt = f f e- P - ' r d s dt 0 0 0 0 0
where p (t) = 2x sinh t cosh s + 2~ t
and q (t) = 2 y sinh t cosh s + 2fl t.
Substituting this in (7.1) and separating real and imaginary
parts, we get
oooo
xo~ x - yfl, = 2lzv'12 f f e -Pcosqdtds 0 0
and oooo
xfl. + y~. = 2lzv'12 f f e-Psinqdtds 0 0
where we changed the order of integration. An application of
Lemmas 4.2 and 2.t to the integrands then yields the result.
These inequalities can be expressed more simply in terms of the
partial derivatives with respect to the polar coordinates r and ~0
of the z-plane, if we make use of the Cauchy-Riemann equations for
the analytic function v (z).
Corollary 7.6. Let H i (z) = 0 with x, y, a, fl positive. Then
~cc < 0 and off > O. ~ ~r
t then ~ ~fl If in addition ~ >-- -- 2 ' -br > 0 and ~
> 0.
8. Behavior of the large zeros
In this section we shall approximate the large zeros by
considering the be- havior of H i (z) for ]vl large compared to I
and ]z].
The first Hankel function may be defined by
Z e - i~
Hi (z) ---- ]-~ ( ) - J~ (z) i sin v It follows that
where
D(v) = v~ (--t)kz*kk! k : l
and
-- ivs inv~H~(2z) = A(v) + D(v)
Z--v e -- i ~ r~ zv
A(v)-- F(--v) + F(v)
Z -v e-- i v~ Z v ] F( - -v+k+l ) + F~k+l ) =--Dl(v) +
D~(v)"
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The Zeros of the Hankel Function 243
Using STIRLING'S approximat ion for the gamma function, we can
approximate the large zeros of A(v). Then we can show that IA(v)l
>[D(v)] on a circle of
radius [lo ~ ]vl ]-1 centered at each zero. By RoucHs Theorem,
the zeros of
A(v) approximate those of H i(2z). These results are summarized
in
Theorem 8.1. Let z=re ~ and 0 t), the approximate expression
for the n-th zero v,~ of He (x), where for large n
and where
and so on.
z o = t .856 e i'fl3
T 1 = 3.245 ei'~l a
9. Survey of results on the zeros o f / /~(x ) , x> 0
Let x be a f ixed posit ive number. Let v=m+i f l be a complex
number and let H i (x) be the Hankel function of the first k ind
and of order v with argument x. We have :
(t) The function H i (x) is an entire function of v which
vanishes for inf initely many values of v= 4-v k, k = t, 2, 3 . . .
.
(2) ~,v=/~ H i (x) = H0 ~ ( , ) ~ - - ~ ,
where the v~ depend on x.
(3) For values Ivl which are sufficiently large compared to I
and to x, the zeros of Hi (x ) are given by v~= -4-(~+ifln)
where
,o o ,,
and where n is a (large) posit ive integer.
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244 WILHELM MAGNUS and LEON KOTIN : The Zeros of the Hankel
Function
(4) For the zeros v~ of H i(x), with k : t, 2, 3 . . . . . we
have
oo
oo " f t 2 K 0 (2x sinh t) dt Re~, I _ o
fK 0 (2x sinh t) dt 0
where K o denotes the modified Hankel function of order
zero.
(5) The following inequalities hold: If v=o~+ifl and H~l)(ar)=O,
where a is real and positive and f l>0, then a0, then
a) ~f l>0,
b) >x-Vx , c) x~f l tan~+ 88 when loci< 89
d) Ifl] > x when ]~ l - - 1 2 2 '
H~(x) ] '
f) l im v : 0. X---> 0
(6) If v=c~+if l and H~l)(x)=0, then ~, fl may be considered as
functions of the positive real variable x. We have
d~ >A I d~ >0 whenever ~= 8 or x>- - .
= 4 d~ dx >0 whenever c r and x>0.
lim d~ +idxf l =oo . x--~O dx
dfl < 2/~ when ~ > t . = Vs -I s
References [1] MAGNUS, W., and F. OBERHETTINGER: Formulas and
Theorems for the Functions
of Mathematical Physics. New York: Chelsea Publ. Co. 1949. ~2]
TITCHMARSH, E. C.: The Theory of Functions, 2nd Ed. New York:
Oxford Univ.
Press 1939. [3] WATSON, G. N. : A Treatise on the Theory of
Bessel Functions, 2nd Ed. New York :
Cambridge Univ. Press 1944.
New York University 25 Waverly Place
New York 3, New York and
U. S. Army Signal Research and Development Laboratory Fort
Monmouth, New Jersey
(Received November 9, 1959)