[ 811 ]
TWO INEQUALITIES FOR PARABOLIC CYLINDER FUNCTIONS
B Y F. W. J. OLVER
Communicated by E. T.
GOODWIN
Received 1 October 1960
1. Introduction and summary. In this paper upper bounds are established for the
principal solution of the differential equation
d
^=fi^-\)w
(1-1)
and its derivative, for unrestricted values of the complex variable t and the complex
parameter /*. The results may have little interest in their own right, but they are of
great value in developing the asymptotic theory of linear second-order differential
equations in a domain containing two turning points. Equation (1-1) is the simplest
example of a differential equation of this type.
An upper bound for a solution of (1-1) has previously been obtained by Watson (5)
in the special case when /t2 is an odd positive integer, by majorizing an integral representation. For these values of ji the solution can be expressed as a Hermite polynomial.
We use here a different approach, applying results recently obtained by the present
writer (4) concerning the uniform asymptotic behaviour of solutions of (1 • 1) for large \/i |.
This approach yields a more powerful type of inequality and illustrates how the
structure of the majorizing function is related to the actual behaviour of the parabolic
cylinder function.
In the next section some preliminary definitions are given. The main results are
presented as a theorem in § 3, and proved in §§ 4-6. In the concluding section, § 7,
a brief examination is made of the limiting forms of the present inequality and the
inequality of Watson for large values of fi and \t\.
2. Definitions. We use the notation of Miller (2) for the parabolic cylinder functions.
The standard form of differential equation for these functions is then given by
g ? = (!*« +a) w,
(2-1)
which is transformed into (1-1) by the substitutions
a = -\i&,
z = /dj2.
(2-2)
The principal solution, U(a,z), of (2-1) is determined by the condition
U(a, z) ~ z-a~i e~iz° as
z^-+oo.
(2-3)
In the notation of Whittaker (6) for the parabolic cylinder function Dn{z) and the
confluent hypergeometricfunction Wkm(z), we have
U(a, z) = D_a_|(z) = 2-i°z-iW_ia,_i($z*).
52
(2-4)
Cambs. Philos. 57, 4
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812
F. W. J. OLVER
In constructing the desired inequalities for U( — %fi2,/itj2) and its derivative
U'( — %/i2, (it^]2), we shall use the function
£(0 = C{t2-l)tdt=it(t2-l)l-$ln{t+{t2-l)i}.
(2-5)
On expansion, we have
The function £,(t) has branch points at t = + 1 and infinity. The branch we use depends
Q= a r g / t ;
on 6, where
andweshalldistinguishitbyusingthenotationg fl (i),reserving^(i) for the many-valued
function. The branch Ee(t) is determined as follows. Let the points of affixes + 1 and — 1
in the £-plane be denoted by A and E, respectively. From each of these points there
emanate three curves along which
gt{eaw£(*)} = constant.
(2-7)
We call these curves the principal curves; they have tangential inclinations to the
positive real axis of + %n - f6 and n - %6 at t = 1, and ± § n — f #, — f # at t = — 1. For
large \t\, the six curves are asymptotic to the four rays arg£ = + \n — 6, + \~n — 6\ this
follows from (2-6).
Fig. 1 illustrates the principal curves when 0 < 6 ^ IT. Those through A are the
curves AP, AQ and AS when 0 < 6 < \TT, and AP, AQ and AR when ^n < d < n. When
6 = 0, AQ degenerates into the curve EQ plus the join of A and E; similar degeneracies
occur when 6 = \i\ and n. The principal curves emanating from E are the images in the
origin of those through A; this can be verified by use of the formula
(2-8)
£(*e±<») = £(*) + **»,
obtained from (2-6). The closed domains into which the <-plane is divided by the
principal curves are denoted by \Jj(6) (j = 0,1, 2, 3,4), as indicated in Fig. 1.
We can now define £0(t). When 0 ^ 6 ^ n, we cut the t-plane along the curves AQ and
ER, indicated by the thickened lines of Fig. 1, and select the branch which is continuous
in the cut plane and for which
£e(0) = %ni (0<6<n),
^ ( 0 - 0 i ) = \ni (6 = 0,n).
(2-9)
When — n ^ 6 ^ 0, the cuts are taken to be the conjugates of those for — 6, and the
branch isfixedby
£„(()) = -i7rc ( - 7 r < 0 < O ) ,
£g(0 + 0i) = -hni (6 = 0, -n).
(2-10)
That (2-9) and (2-10) are consistent when (9 = 0 follows from (2-8).
This defines £,g(t) everywhere in the <-plane, except in the region, \J2(6), to the left of
EQ + ER when 6 = 0, and to the right of AQ + AR when 6 = ± n. We complete the
definition by supposing that
&(*) = - & ( - < )
(0 = 0),
g«(0 = ± * « - & ( - * )
{6 = ±n),
(2-11)
(2-12)
when t e U2(0).
It may be noted that when t is real and exceeds unity, £,g(t) is real and positive
provided that \6\ < fn. If fn < 6 < TT this is no longer true, because the cut AQ (which
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Two inequalities for parabolic cylinder functions
813
coincides with the real axis when 6 = §77) then lies on the opposite side of the real axis.
Similarly when — fn > 6 > — n.
Except for the use of the subscript d on £ and the addition of (2-11) and (2-12), the
notations and definitions of this section are the same as those introduced in (4), §§ 3, 5.
Fig. 1. Principal curves and domains Vt(d).
3. Main results.
THEOREM.
When |arg/i| < n
(3-1)
• (3-2)
52-2
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814
F. W. J. OLVER
Here E,e(t) is the function defined in § 2, (2e)~i'j!! and /d^2 have their principal values,
and k is a positive number independent of ji and t. An actual value for k has not been
determined, nor is it required in the applications contemplated in § 1. In the following
sections we shall use the symbol k generically to denote any number which is independent of fi and t.
A proof of the theorem is given in the next three sections. In §§ 4, 5 we establish (3-1)
and (3-2) for |/t| ^ M, where M is an assignable constant; the extension to \/i\ < M
follows in § 6.
4. Proof for large \/i\. From<4), (8-11) and (6-2), we have
^ & l | a & > ] ,4-1)
as \/i\ -> oo, uniformly with respect to t in a domain T(8), defined below, and 6 in the
interval (— 77+ e, n — e), where e is an arbitrary positive number. Here g(/i) is given
asymptotically by
(4-2)
%
and the function £e = £e(t)- is defined by
§&(*)}* = £>(*);
(4-3)
£g(t) has a branch point at t = — 1, but unlike E,${t) it is analytic at t = 1. The only cut
associated with £g(t) is the principal curve ER of Fig. 1 if 0 < 6 ^ n, or the curve conjugate to ER if 0 > 6 Ss -77, or EQ + ER if 6 = 0. For all values of 0 in the interval
— n^6^
77, £e(£) is real and positive when t > 1. The functions {^/(<2— l)}i,As(£e) and
B8(£e) are regular in the same cut plane, and if a neighbourhood of t = — 1 is excluded
A8((,g) and (1 + |^|*)£ s (^) are bounded with respect to t and 6.
The domain of validity T(#) is a subregion of the cub plane, and is illustrated in
Fig. 7 of (4) when 0 < 6 ^ 77 — e. It is defined as
{\n-e
^ 6 ^ 77-e), (4-4)
where V3- = V^(0) is the region obtained by adding to U3-(#) points whose £-maps are at
a distance less than S from the £-maps of the boundaries of Uj(6), S being an assignable
positive constant depending on e, and where ^V3- is the region complementary to V^
(compare (4), §5).
From (3), §4, we have the Airy-function inequalities
|A|i)|exp(-|A*)|,
(4-5)
where A is an arbitrary complex number, and fAt takes its principal value. If we set
A = fii£g(t), then arg A can take the values ± 77 only on the principal curve AQ when
0 ^ 6 < 77; this is clear from Fig. 6 of (4), in the notation of which A = \fi\$z. Hence in
T(d) the principal value of fAt is fi2£g(t). Since T(#) excludes a fixed neighbourhood of
t = — 1, we see that when \fi\ ^ M, assignable, the content of the square brackets in
t\erp(-^)\
.. „
(4-1) is bounded by
(
}
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Two inequalities for parabolic cylinder functions
815
Substituting (4-2) and (4-6) in (4-1), we derive
-\/l2,[lt^2)\
i|exp(-y
•
(4-7)
From (2-6) and (4-3), we see that
as |<| ->- oo, uniformly with respect to argi. Hence
(4>9)
and since £fl/(<2 — 1) a n d its reciprocal are regular functions of t in T(0), we derive the
inequalities
,
,„ ,i
,
in which, of course, the two values of the positive generic constant k are not the same.
On substituting for \£g\i in (4-7) by means of these two inequalities, we establish (3-1)
with the conditions
. „,,,. ,ai „
, , _ „,
.. . . .
JeT(0) \d\^TT-e, \/i\^M.
(4-11)
By use of (4), (8-15), we may verify in a similar way that (3-2) holds in the same circumstances.
The expansion (4-1) is one of two main types which were given in (4) for large |/t|.
The other type involves only elementary functions; corresponding to (4-1) we have
U(-w,fitV2)~fa)ex^2^!e)j0^
M
(see (4), (4-3)), where the s#s{E,B) are bounded. The connexion of (3-1) with this expansion is more obvious than that of (4-1). The region of validity of (4-12), however, does
not include the turning points t = ± 1, and it is not possible to determine a majorizing
function for U( — \/i2, fit^2) in regions which include these points by use of (4-12).
5. Proof for large \/i\ {continued). In this section we extend the result just obtained
to the ^-region complementary to T(#), and to the full range ( — n,n) for 6.
In (3-1) let us replace ji by /ie-ni and t by — t. We obtain
and applying the same substitution to (4-11), we see that (5-1) holds with the conditions!
t€-T(6-n),
e s£ 0 < 2n-e,
\/i\ > M.
(5-2)
With the aid of (2-7) and Fig. 1, we can verify that when 0 < 6 < n the cuts in the
i-plane for the function £#_„( — t) are the same as those for £,g{t), namely the principal
curves AQ and ER. Now
(<2-l)*d* = - f (t*-l)idt,
£eJ-t)=[
Ji
(5-3)
J-i
•(• By — T(#—77) we mean the image in the origin of T(0 — n).
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816
F . W. J . O L V B R
where the branch of (t2 — 1 )i is the one which is a positive multiple of i when —
The branch of (t2 — 1)£ in the defining relation
\<t<\.
has the opposite sign. Hence in terms of the latter branch we have
(0<0<n).
(5-4)
The substitution of (5-4) in (5-1) reproduces (3-1). If e < # ^ TT —e the union of the
domains T(#) and — T{6 — n) comprises the whole f-plane. Hence combining (4-11)
with (5-2), we see that (3-1) holds with, the conditions
tunrestricted,
e < 6 ^ IT — e,
\/i\ ^ M.
(5-5)
In a similar way, we can show that (3-2) holds in these circumstances.
The above analysis does not apply when 0 < 6 < e; in order to achieve the desired
extension for this range of 6, we use the connexion formula
obtained from (5-26) of (4) by changing the signs of i and t.
Now when 0 < 8 ^ e and \/i\ 5= M, we find that
cos
k |exp ( - inifi2) exp (-1^«) $fi2)W\,
(5-7)
on using (4), (2-22). In (5-6), we substitute the inequality (5-7) and the two inequalities
obtained from (3-1) by (i) changing the sign of t, (ii) replacing /i by [ie~^ni. We obtain
and from (4-11) and the conjugate of (5-5) we see that this result is proved when
t e -T(6),
O^d^e,
\ii\>M.
(5-9)
The cuts for the branches £e( — t) and £e_j,,(£), and also for £g(t), are illustrated in
Fig. 2 when 0 < 6 ^ e. From (2-5) we have
U-t)
= f~ (t2-l)idt = - f («»-l)*cB.
(5-10)
Both integrands have the same branch as that of (2-5) when — 1 < t < 1, and therefore
also when flies in Uo(#), U2(0) or U4(0); hence
(*€U O (0)UU 2 (0)UU 4 (0)). (5-11)
The effect of crossing A Q is to change the sign of £g(t)
(5-12)
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Two inequalities for parabolic cylinder functions
817
and the effect of crossing ES is to change the sign of E,g{ — t)
ie(-t) = -\ni + gg{t) (te\J3(d)).
(5-13)
In consequence of these three equations we may assert that
l
m
i
l
(5-14)
For, when t e U2 U U4, fit{/i%g(t)} is non-positive (see Fig. 3 of (4)), and (5-14) follows
from (5-11); when< e U1; (5-12) immediately establishes (5-14); and when t e U3, (5-14)
follows from (5-13) and the fact that \e^"l\ < 1. Thus (5-14) is proved when 0 < 6 =* e.
When 6 = 0, equations (5-12) and (5-13) again hold, but in place of (5-11) we have (2-11)
when t e U2(0). It is easily seen, however, that (5-14) again holds.
••••p
Fig. 2. <-plane, 0 < 0 s£ e. (a) Cuts
(6) Cuts
for E,g(t);
for ge(t);
for £g(-t).
for ge_±n(t).
In a similar way, we derive
(5-15)
(5-16)
and thence
|exp (/J,%_h(t))\ < |exp (-/i%(t))\
(t e «"UO(0)).
(5-17)
Substitution of (5-14) and (5-17) in (5-8) again reproduces (3-1). Since the domain
— T(6) is contained in tfU^d) when 0 ^ 6 ^ e, we have shown that (3-1) is valid with
the conditions (5-9), and similarly also when the second condition is replaced by
0 ^ 6 ^ — e. The union of — T(d) and T(#) comprises the whole <-plane; we have
therefore established (3-1) with the conditions
tunrestricted,
\6\ < e, \/i\ > M.
(5-18)
Similarly, we may show that (3-2) holds in these circumstances.
It is perhaps worth remarking that the more elaborate proof required for the extension to the set of conditions (5-18) as compared with that for (5-5), may be regarded as
a consequence of the greater complexity of the asymptotic behaviour of U( — %fi2, fit*j2)
for large |/t| when \9\ < e. When/*2 = 1,3,5,..., this function is exponentially small in
the region U2(#), whereas for all other values of ji it is exponentially large in V2(6).
Thus, in contrast to other regions of the £-plane, U( — \p?, /it *J2) cannot be expanded
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818
F. W. J . OLVEK
uniformly in U2(0) in terms of a single Airy function (compare (4), § 9). As a further
observation, the inequalities (3-1) and (3-2) are far from sharp in character when
/i2 = 1,3,5,... and t lies in U2(#), because the factor |exp(— (i2£g(t))\ becomes
exponentially large as \/i\ -*• oo.
The final extension we make in this section is to ^T(d) when n — e ^ \6\ ^ n. To this
end, we replace /i by ft e~ni and t by — t in (3-1), again leading to (5-1). In consequence
of (5-18), this inequality is therefore valid with the conditions
t unrestricted, v-e^d^n
+ e, \fi\^M.
(5-19)
The branch £#_„( — <) is expressed in terms of £,g(t) by (5-4) when 0 < 8 < n, and also
when 6 = n provided that «€^U 0 (0); whence U0(0) we deduce from (2-11), (2-12) and
(5-4) that E>0{ — t) = £„(£)• Using these results, we deduce that (3-1) is valid with the
conditions
\/i\^M.
(5-20)
f unrestricted, n-e^e^n,
Similarly for (3-2).
Combining the conditions (5-5), (5-18), (5-20) and their conjugates, we have shown
that the inequalities (3-1) and (3-2) hold with the conditions
\6\ =g n,
tunrestricted,
\/i\ Js M.
(5-21)
6. Proof for bounded \/i\. We require the following asymptotic property of the
principal parabolic cylinder function
a+
*J ^ + * )(a+ | ) ^ + * )(a + *
)
}
(6-1)
as \z\ -*• oo, uniformly with respect to a in any bounded region and argz in the interval
(-|77- + e, |?7-e).
The validity of (6-1) for fixed a has been established in (6), § 16-5, the proof of its
uniformity with respect to bounded a, which appears to be new, may be achieved with
the aid of the Barnes-type contour integral ((2), equation (4-71); (6), § 16-4)
which is valid when a 4= — \, — f, — f, • •. and |arg z\ < fn; the contour of integration
separates the poles of T(s) from those of V(% + a — 2s), and (z^/2)28 takes its principal
value. By translating the contour a distance N to the left, where N is an integer, we
n=0
where
fl
= * f ~W+1 * v " ; l ) ' " " ~'(zJ2)*ds.
^TT* J _jv_i 00
(6-4)
1 (^ + a )
Now suppose that JV exceeds the upper bound of |£a|. Then all the poles of F(£ + a — 2s)
lie to the right of the line SRs = — N + \, hence the contour in (6-4) may be deformed
into this line. Considerations of continuity now show that, unlike (6-2), the equations (6-3) and (6-4) are valid for all values of a in the circle \a\ < 2N — 8, where 8 is an
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Two inequalities for parabolic cylinder functions
819
arbitrary positive number, including the values —\, — f,..., — (2N — £). Taking a new
variable of integration s' = (s + N — \)ji and applying Stirling's formula for the gamma
function ((l), §9-55), we readily show that under the assumed conditions
\RN\ < k\z\-W-*\
(6-5)
and thence establish (6-1).
The asymptotic behaviour of U(a, z) for large \z\ in other phase ranges may be found
with the aid of connexion formulae. From (2), equation (32), we have
U(a, - z) = exp {ni(a + £)} U(a, z) + e x P M | a - i ) } ( ^ ) *
U{_a>_
- z)
(6.6)
Substituting by means of (6-1), we obtain
U(a, -z) = ex
where the terms 0(|z|~2) are uniform with respect to bounded a, and with respect to
arg z in the interval (— %n + e, §n — e).
By setting a = — \fiP, z = jit ^2 in (6-1), we derive the inequality
\U(-l/i*,tdj2)\ <k\(/j,tp)i^-iexV(-^H%
valid when
\/i\ ^ M,
(6-8)
|arg(/t*)| < %-n-e, \fil\^P,
(6-9)
where M is the number introduced in § 4, and P is assignable. Application of Cauchy's
integral formula for derivatives ((l), § 4-32) shows that formal differentiation of (6-1) is
valid. Hence, with the same conditions,
| £ 7 ' ( - ^ , / ^ V 2 ) | <&|(/rfV 2 )*" 2+ *exp(-£/^ 2 )|.
(6-10)
Similarly, from (6-7) and its conjugate, we derive
^ *
(6-11)
(6-12)
valid when
\/i\ ^ M, %n + e < |arg(/t«)| < t^-e> \pt\ > P.
(6-13)
Comparing (6-8) and (6-11) with (3-1), and (6-10) and (6-12) with (3-2), we deduce
that (3-1) and (3-2) are valid with the conditions
\d\ < n,
\/i\ < M,
\/d\ > P,
(6-14)
provided we can show that
|Z|, 1/|Z|, 1^1 < k (\6\ < 77,
and
where
|r,| < k (\0\ < n,
\/i\ ^ M,
H ^ M,
\/d\ > P),
f w - e < |arg(/*«)| ^ n,
\/it\ > P),
(6-15)
(6-16)
X s {jd J2)-*{1 + [p\*+ \/t\* \t2- l|i},
Yt = (fit p)^ exp ( - £/^2) exp (v%(t)),
(6-17)
(6-18)
Y2 = ( ^ V 2 ) - * / ' s e x p ( ^ 2 ) e x p ( / t 2 ^ ) ) )
(6-19)
the powers of fit ^/2 having their principal values.
We now establish (6-15) and (6-16). In view of symmetry, it is sufficient to consider
the ^-interval 0 < d < n.
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820
F. W. J. OLVER
The conditions of (6-15) and (6-16) both imply that
|*| Ss P/M = T, say.
(6-20)
The boundedness of |X| and its reciprocal is a simple consequence of this result and the
conditions of (6-15).
As \t\ ->• oo in the union of the domains Uo, U^ U3 and U4 1; where U4 1 denotes the
part of U4 which lies below the real axis, we see from (2-6) and Fig. 1 that
£&) = \t*-\\ng2t-\
+ 0{\t\-*),
(6-21)
where \ng 2t is the branch of the logarithmic function which is continuous in the £-plane
cut along the principal curves AQ and ER, and takes its principal value at the point P.
Therefore
);
(6-22)
here (and in the rest of this section) In denotes the principal value of the logarithmic
function, and j \ = 0 or 1. The substitution of (6-22) in (6-18) shows immediately that
Fx is bounded for these values of t.
In the remaining part of the i-plane, namely U2 U U4 2, where U4 2 is the upper part
of U 4 , we have
&(0 i ? > » - & ( - 0 ,
(6-23)
where j a = 0 or 1, depending whether d is zero; see (5-11), (2-11) and (2-12). Since
— t e U 0 U U 4 1 , we derive from (6-22)
ge(-t) =
tf*-l]n(-2t)+j1m-±+0{\t\-*).
(6-24)
With the aid of (6-23), (6-24) and (6-18) we readily show that exp{2/i2£,g( -1)}Yx is
bounded. From Fig. 3 of (4) and the fact that —t e Uo U U 4jl , we see that
Hence |exp{— 2/i2E,e( —1)}\ < e"1^1*, which is bounded, and therefore Y± is bounded.
This completes the proof of (6-15).
Next, consider Y2. From Fig. 1, we see that in consequence of the condition
| 7 r _ e «- |arg(/rf)| < n> t cannot lie in Uo. When k U ^ U j U U 41 , Equation (6-22)
holds, and with its aid we easily show that exp { — 2/i2£g(t)} Y2 is bounded and therefore,
since $Re{/t2£e(£)} < 2n |/*|2, that F2 itself is bounded. In the remaining region, U 2 U U4 2,
we establish the same result by substituting (6-23) and (6-24) in (6-19). This completes
the proof of (6-16).
The master inequalities (3-1) and (3-2) have now been established with the conditions
(5-21) and (6-14). The only combination of variables which remains to be considered is
therefore
|0| s£ 77-, \/i\ ^ M, \id\ < P.
(6-25)
From the series expansions given on pages 62 and 63 of (2), we may show that U(a, z)
and U'(a, z) are bounded when a and z are bounded. Hence, with the conditions (6-25)
we have
\ U \ \ l \
k
\
U
p
\
k
.
(6-26)
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Two inequalities for parabolic cylinder functions
t
2
821
±1
Since (2e)i'' /i'^ (I + |/*|*+ M* l* ~ l|^) is clearly bounded in the present circumstances, it remains to show that exp {/j?£g(t)} is bounded.
In the complement of Uo this result is an immediate consequence of the inequality
9K/*2£e(0} ^ i77!/*!2- The result is also obviously true in any bounded sub-region of Uo,
say \t\ ^ T. In the part of U o lying outside this circle we have, from (6-22),
fi%(t) = %flH2-^]n2t
+ fi*{j1Tri-l + O(\t\-%
7
(6-27)
2
and since \t\ lies between the limits? and Pl\fi\, it is clear that |/t ^(<)| is bounded,and
therefore also |exp{/t2gfl(<)}|.
The proof of the theorem of § 3 is now complete.
7. Watson's result. In § 9 of (5), Watson proves that when n is a positive integer and
z is unrestricted
where
|Z)m(2)| < | * J 2 ) | ,
(7-1)
®n(z) = T(n + l)eln(2n)-n {z + (z2 - 4n)i}n exp { - \z(z2 - 4w)*},
(7-2)
2
the branch of (z — 4n)* being determined by
| Z - ( Z 2 - 4 T I ) 4 | ^ 2Vn.
(7-3)
In the present notation, we have
Dn(z)=U(-^,/j,tj2),
n = ^-l,
z = /itj2.
(7-4)
Watson remarks that a similar result may be established by his method when n is not
restricted to be an integer, but the extension is not trivial.
Compared with (3 • 1), the inequality (7 • 1) has the merit that it contains no unspecified
constant k. We shall now demonstrate, however, that the character of (3-1) is the
stronger by considering the limiting forms for large n (or fi), and large \t\.
(i) Limiting forms as /i->- + oo, t being bounded away from the turning points. Let t lie
in the domain S(0) illustrated in Fig. 2 (a) of (4), so that l/\t2- 1| is bounded. Then
the branch of (t2— 1)* here being the principal one, in consequence of (7-3), and the
0 being uniform with respect to t. Substituting the limiting forms of the various
functions on the right of (7-2), we obtain, after some calculation,
<Dm(2) ~ 2i7r*(2c)-i''V^2+i{< + (t2-1)*}^2-4exp{-
\/iH{t2- 1)*}.
(7-6)
The actual behaviour of the parabolic cylinder function in these circumstances is
given by
U
{
. ^ ^
2
) ~ 2 - i ( 2 e ) - ^ > ^ - * ( ^ 2 - l)-iexp{-/*%,(*)};
(7-7)
see (4), (4-3) and (6-2). Substituting (2-5) for £,g{t) and dividing the right of (7-6) by that
of (7-7), we find that the inequality (7-1) overestimates the actual behaviour of the
parabolic cylinder function by the factor
which tends to infinity with /i. On the other hand, the corresponding overestimation
factor for the limiting form of (3-1) is 2$k and is independent of the variables.
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822
F . W. J .
OLVER
(ii) Limiting forms as /i-^ + co when t = 1. In this event
®n(z) ~ 2M(2e)-^V^ 2 +*
(7-8)
(compare (7-6)), whereas from (4), (8-11) and (6-2), we find
U( - &*, ft V2) ~ 2A77i(2e)-^V^-4/{3?r(f)}.
(7-9)
The overestimation factor for (7-1) is therefore
which is again an unbounded function of /*, whereas the corresponding factor for the
limiting form of (3-1) is the constant
(iii) Limiting forms as \t\ -» oo; ju, bounded. In this case we find that
(7-10)
Comparing this with (6-1), we deduce that when |argz| < £TT — e the overestimation
factor for (7-1) is
The corresponding factor for the limiting form of (3-1) is 2iJc.
The work described above has been carried out as part of the research programme of
the National Physical Laboratory and this paper is published by permission of the
Director of the Laboratory.
REFERENCES
(1) COPSON, E. T. Theory of functions of a complex variable (Oxford, 1935).
(2) MILLER, J. C. P. Tables of Weber parabolic cylinder functions (London: H.M. StationeryOffice, 1955).
(3) OLVBB, F. W. J. The asymptotic solution of linear differential equations of the second order
for large values of a parameter. Phil. Trans. A, 247 (1954), 307-27.
(4) OLVBB, F. W. J. Uniform asymptotic expansions for Weber parabolic cylinder functions of
large orders. J. Res. Nat. Bur. Stand. 63B (1959), 131-69.
(5) WATSON, G. N. The harmonic functions associated with the parabolic cylinder. Proc. Lond.
Math. Soc. 17 (1918), 116-48.
(6) WHITTAKER, E. T. and WATSON, G. N. A course of modern analysis, 4th ed. (Cambridge, 1927).
T H E NATIONAL PHYSICAL LABORATOKY
TEDDINGTON
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