An asymptotic expansion for the high-frequency superposed dynamic viscosity of the Doi-Edwards fluid

Im. 1. Non-Lmcar Mechanics. Printed in Great Britain.

Vol.

27, NO. 2. PP. 29S308.

0020-746?1’92 s5.00 + .oo Pergamon Press pk

1992

AN ASYMPTOTIC EXPANSION FOR THE HIGHFREQUENCY SUPERPOSED DYNAMIC VISCOSITY OF THE DOI-EDWARDS FLUID B. BERNSTEIN Department of Mathematics, Illinois Institute of Technology, Chicago, IL 60616, U.S.A. and

R. R. HUILGOL School of Information Science and Technology, The Rinders University of South Australia, G.P.O. Box 2100, Adelaide, SA 5001, Australia (Received 14 October 1990; accepted

1.3 December

1990)

Abstract-It has been observed that when small oscillations are superposed on a steady shearing flow of a polymer fluid, the superposed dynamic viscosity becomes independent of the underlying steady shear rate at high frequencies of oscillation. Previous predictions of such behavior from the BKZ theory depended on the assumption that the linear shear modulus, G(t), can be continued analyti~IIy across the ima~nary axis in the complex r-pfane. It is shown that this cannot be done for a Doi-Edwards fluid and, therefore, that the previous arguments do not apply to this fluid. Nevertheless, a new derivation establishes that the observed high-frequency behavior of the superposed dynamic viscosity is, indeed, predicted by the Doi-Edwards fluid. An asymptotic expansion of the dynamic viscosity is ako obtained.

I. INTRODUCTION

interesting phenomenon has been seen when a small, shearing oscillation is imposed on a large, steady shearing motion of a polymeric fluid. It has been observed that at large frequencies of oscillation, the superposed dynamic viscosity tends to become independent of the underlying rate of steady shear. Booij [l] found such a tendency in cone and plate flow, in which oscillations were imposed in-line with the steady shear. Simmons [Z] observed the disappearance at high frequencies of the dependence of the dynamic viscosity on the steady shear rates when, in1968, he superposed small oscillations transverse to the steady shear in a flow between concentric cylinders by oscillating the inner cylinder in its axial direction while the outer cylinder was turning at a steady rate. In reporting on their repetition of Simmons’ experiment and on their concurrence with his observation, Tanner and Williams [3] offered an argument that the BKZ [4,5] theory predicted the independence of the underlying steady shear of the superposed dynamic viscosity in the limiting case of high frequencies. In improving their argument, Bernstein and Huilgol [6] showed that the BKZ theory predicts this result under conditions that the linear shear relaxation modulus has finite derivatives at t = 0. They also showed that the general simple fluid is not in agreement with this result. Bernstein et al. [7] then obtained an asymptotic series for the superposed dynamic viscosity of a general BKZ fluid for which G(t) is well behaved at t = 0. Answering an objection by MacDonald [8], Bernstein and Huilgol [9] showed that, indeed, the result could be extended to the situation where d(t) becomes infinite like t” as t -+ 0 and m > - 1,provided that Watson’s lemma could be used.’ However, as we shall demonstrate here, amplifying the summary that Bernstein [lo] presented at the X International Congress on Rheology, the result in [9] does hold for the Doi-Edwards fluid [ll-141, even though none of the earlier arguments is valid for such a fluid. An

2. DYNAMIC

MODULI

FOR THE DOI-EDWARDS

FLUID

In 1971, Bernstein and Huilgol [6] obtained expressions for the superposed dynamic viscosity for the BKZ fluid. For an oscillation of frequency w superposed on a steady flow of ‘In a thesis [lS] written under the joint supervision of the authors, Rassaian showed that our results were implied by Watson’s lemma. 299

300

B. BERMTEIN and R. R. HUILGOL

shear rate G:(K,

w),

K, the transverse dynamic modulus, are given by

G/‘(K,o)

= G; + iG;’ =

G:(K,

r

and the in-line dynamic modulus,

w)

CJ(KS,S)(~

e-‘““)ds

(2.1)

- e-‘“‘)ds

(2.2)

-

s0 and G:(K,o) = Gi + iG;’ = respectively, where

I

oSf(~s,s)(l

.

/(?94 =

4-m. 4 = 9th s) + 72

(2.3)

for arbitrary 7 and s, and where g(y,s) figures into a general shearing motion, for which the amount of shear, ;.(t), is an arbitrary function of r, as follows: The shearing stress, c at time t is given by

a(t) =

I

s -*

gb(t) - Y(S),t - 51b(t) - ~(41 d7.

(2.4)

Comparison of equation (2.4) with the corresponding expression for a DE (Doi-Edwards) fluid [I l-141 leads to the following relations for such a fluid

g(i’,4 = GorptA~‘(s)

(2.5)

fb 4 = Go@(Y)P’(s) where Go is a constant and ./2x2 - 1 /(.12x2 - 1)2 +

1

dx

4px4

(2.6) (2.7)

where Td is a material constant known as the disengagement rime. If we consider equation (2.4) for infinitesimal strain, we obtain, with the use of equation (2.5) G(t) = - g(O,t) = -f(O,t)

= - Go~(0)$(t)

(2.9)

where G(r) is the linear relaxation modulus. Equation (2.9) will be useful for comparison with classical linear results. The transverse and in-line superposed dynamic viscosities, $(K, o) and ?,I;(K, w), are given respectively by

(2.10) v;(K,

()))

=

3

(G:>- Gf’tK,al 0

0

Let us now ask what the results of Bernstein and Huilgol [9] would have to say about the limits of the dynamic viscosities for a DE fluid. But first, we must seek the behavior of G(t) as t + 0. That ~(0) = l/5 is shown in Appendix E, whence equations (2.5), (2.7), (2.8) and (2.9) give -

G(r)= 2SrrgT ,zd e-p2tlTd.

(2.11)

It is shown in Appendix A that - lim &G(t) t-0

=

2Go S.J;TTd

(2.12)

Dynamic viscosity of the Doi-Edwards

301

fluid

which implies that G(t) = 0(t- l”). Were the arguments of Bernstein and Huilgol [9] applicable, we would obtain, in their notation, lim w~!~~&(K 9w) = lim w3”$(k I ?0) = const. 0-x w-3

(2.13)

However, in deriving equation (2.13), Bernstein and Huilgoi [9] required that in some complex neighborhood of s = 0 one must have (2.14)

G(s) = s”_&(s)

where m > - 1 and fo(s) is an analytic function of complex s. [Clearly, equation (2.12) implies that m would take the value -l/2 for a DE fluid.] But, a consequence of this requirement is that G(s) can be analytically continued across some interval of the imaginary axis in the complex s-plane. That this is not so for DE, i.e. that the imaginary axis forms a natural boundary to the G(s) of the DE fluid is shown in Appendix B, and thus the argument of Bernstein and Huilgol [9] cannot be used here. Nevertheless, as a consequence of the asymptotic results which we derive here, it will follow that, indeed, equation (2.13) does hold for a DE fluid.

3. EXPANSIONS

FOR

THE

DYNAMIC

MODULI

Let us now put equations (2.5H2.7) into (2.2). It then follows that

=

2

G:(K, cc) -

;c

zl

d podd

0

dfc4 e-(io-Pmfls&

(3.1)

where 8Go G:(K, co) = -

x

(p&s) c

s0

K2Td

e-p’fTd ds

(3.2)

podd

and where the interchange of summation and integration is justified in Appendix D. If we let K = Tdti,

R=

TdW

(3.3)

we obtain from equation (3.1) through a change of variable of integration, C = s/T,+, cp(KC)e- ~P~+iW(j<*

G:(K,o) = G:(K, co) - 3

(3.4)

For G:, we get similarly G~(K,w} = G~(K, co) -

F

1 pcdd

= G~(K,

a~)--$

G:(K, co) =

Jo= ’

Kscp’(Ks) + ~~$Ks)]e-‘~‘+“~“ds

p:d

where

w(;lf(Ks)e-(P’+inffds

I 0

J

[KS(P’(KS)+ (P(KS)] c 1”; om

zd

(3.5)

e-p”rdds.

(3.6)

podd

We shall now turn to the asymptotic expressions at infinite frequency.

4. ASYMPTOTIC

EXPRESSIONS

FOR

THE

DYNAMiC

MODULI

Using integration by parts, we have

J

co

0

cPFs)e

-(p*+inb&

=

2 p

P(O) + - K +iR

p2+iR

J

g

0

Co’VW-(p*

+ ilh

ds.

(4.1)

302

B. BERNSTEW and R. R. HLXGOL

Now since 9(;t) is an even function of 7, it foflows that 9’(O) = 0, and, indeed, that all derivatives of odd order of 9(s) vanish at s = 0. Therefore, a repetition of integration by parts gives for equation (4.1) -(P.‘+inJS&

cp(Ks)e

cp”(Ks)e

-(P: + in’s &

(4.2)

Indeed, continuing to integrate by parts, we obtain for n = 0, 1,2,. . 9(Ks)e

- tp’ + iflJs &

K2”cp”“‘(o)

KZ9”(0)

9(O)

=L

m+(f2+j~)3+“.

+(P2+~~)2n+~

K 2n+2 + (p2

9(2n+2J(KS)e-‘P”+inJSds.

(4.3)

+ isf)2”‘2

We shall apply equation (4.3) to the expression (3.3) for the transverse dynamic modulus. Note also that (j’“’

ii,(j)cj) = pi’&)

= (j + 1)9(j'4)+ p9o’j“‘k;)

d-i

j = 0, 1,2, . ‘ . .

(4.4)

Thus ~(‘~(0) = (j + 1)9(~)(O) whence the expression similar to equation (4.3), but appropriate $(Ks)e

-lP’

+

in)*

&

3K29”(0)

9(O)

_

P2+i~+(p2+i~)3+

to the in-line modulus, is (2n + l)K2”9”“‘(0)

“’

+

(p”+iQ)‘“+l

K2n*2 + (p2

+

iR)2”+2

+ J&p+

3)(Ks)le

n + 3)cpf2”+“(KS) -'Pl+ inJsds.

(4.5)

For G,+ we have, using equations (3.3) and (4.4), G:(R,w) = G:(K, &) -

+ K29”(0) 1

’ podd(p’ + iQ3 + ’ . ’

' podd(p2 + iR)2"+' KZ"+Z cp(2n+2)(Ks)e-( +c podd(p2 +02"'* 1 I- K2”9’2”‘(0) 1

(4.6)

and for Gt we have the same expression with each 9”‘(O) replaced by (k -+ 1)9’“(O) and 9(2n+ “(KS) replaced by +@* 2J(KS). From this, we shall obtain the asymptotic series for G: and also for CT. Indeed, using equations (F.i 1) and (F.15) of Appendix F, we get from equation (4.6) G:(K,o)

- G:(K, 3~) -

(4.7)

and, similarly G:(K,o)-

G:(K,x)--

= ( - 1)“(4n)!(2n + I)&9’2”‘(O) I?” ~(1 [(2n)!]224”

- i)

(4.8)

where the first few terms are calculated with the use of results obtained in Appendix E, namely

9(O)= ; p(o)

= - ;

Dynamic

viscosity

of the Doi-Edwards

fluid

303

40 qp’(O) = 143 qp’(0)

= -

5040

(4.9)

2431’

We shall now examine the limiting behavior of the dynamic viscosity.

5. THE

DYNAMIC

VISCOSITIES

AT INFINITE

FREQUENCY

According to equations (4.8) and (2.6) W3’*‘l:(K,

0)

03’*$(K,

CO)

= ( - 1)“(4n)!@“‘(0)& [(2n)!]224”

L1 -&

so

K*”

(5.1)

2

= (- 1)“(4n)!(2n + 1)(~‘*“‘(0)$ K2n 2’ [(2n)!]22’”

(5.2)

Using equation (4.9), we may display the first four terms of the asymptotic expansions, namely 03’*tj;(K,

cc)) -

6615 o6 K6 + ’ ‘. K4 + --14144 2288 a4

(5.3)

--175

-

and .

(5.4)

In particular, we get for the first-order terms, God In qj - In ~ 57tfi lnql-

ln$!-$

- 31no 2

(5.5)

- ilno

(5.6)

d

so that the DE theory implies that plots of the logarithm of each of the dynamic viscosities vs the logarithm of o at various constant values of K should give curves, all asymptotic to the same straight-line of slope - 3/2, from whose intersection with the axis In o = 0 one may find the value of Go/&. Using the value so obtained, one may then calculate the numerical values of all the other terms in the asymptotic expansions (5.3) and (5.4) and, should good enough data be obtained, verification of these results could provide a check on the validity of the DE equation.

6. CONCLUSIONS

In [lo], Bernstein presented a summary of the results reported here and as promised, we have established the asymptotic series for the superposed dynamic viscosities for the DE constitutive equation and have, indeed, displayed the numerical values of the first four terms of the series. The results agree with those of Booij, Simmons and of Tanner and Williams. We have also shown that these results hold, although the previous analyses of Bernstein and Huilgol do not apply to the DE fluid. The relation found by Bernstein and Huilgol [9] between the nature of the singularity of G(t) at t = 0 and of the dynamic moduli at infinite frequency persists for the DE fluid. It is hoped that the specific numerical results given here can be used to test the validity of the DE equation against experimental data. Acknowledgement-We gratefully acknowledge the support given by The Flinders University of South Australia B. Bernstein as a Visiting Research Fellow in 1988, which made it possible to complete this work.

to

304

B. BERNSTEIN and R. R. HU~LGOL REFERENCES

H. C. Booij. Rheol. Acra 5, 215 (1966). J. M. S. Simmons, Rheol. Acta 7. 184 (1968). R. 1. Tanner and G. Williams, Rheol. Acta 10, 528 (1971). B. Bernstein, E. A. Kearslcy and L. J. Zapas, Trans. Sot. Rheol. 7, 391 (1963). B. Bernstein, rlcra Mech. 2. 329 (1966). B. Bernstein and R. R. Huilgol. Trans. Sot. Rheol. 15, 731 (1971). B. Bernstein. R. R. Huilgol and R. I. Tanner, Inc. J. engng Sci. 10, 263 (1972). I. F. MacDonald, Trans. Sot. Rheol. 18. 299 (1974). B. Bernstein and R. R. Huilgol, Trans. Sot. Rhea!. 18, 583 (1974). B. Bernstein, Proc. X In?. Congr. on Rheol.. Vol. 1, Sydney, pp. 180-181 (1988). M. Doi and S. F. Edwards, J. Chem. Sot. Faraday Trans. II 74, 1789 (1978). M. Doi and S. F. Edwards, J. Chem. Sot. Faraday Trans. II 74, 1802 (1978). hi. Doi and S. F. Edwards, 1. Chem. Sot. Faraday Trans. II 74. 1818 (1978). ,M. Doi and S. F. Edwards, J. Chem. Sot. Faraday Trans. II 75. 32 (1979). M. Rassaian, Asymptotic Relations for Dynamic Moduli, M.S. thesis, Illinois Chicago, Illinois (1974). 16. E. Hiile. Analytic Function Theory, Vol. II. pp. 87-92. Chelsea. New York (1973).

I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

APPENDIX

A: BEHAVIOR

OF

THE

We show here that lim &p’(s) s-0

MEMORY

FUNCTION

Institute

AT ZERO

of Technology.

TIME

2 = ~ l?=,x

(A.1)

and consequently that equation (2.12) holds. To this end, note that for p = 3, 5, 7, . dx _ T,

e -I’a/fr whence, summing all three sides of equation (A.2) over p = 3, 5. 7. resulting sums and then multiplying the three sides by 8,‘n*, we obtain 8 -5-e-“r, n-Td

+

b4.2)

, adding

[l/7,]

x exp( - s/T~) to the

< p’(s)

(A.3)

Now, making

a change

of variable

of integration

to 5 = xm

and multiplying

by 4,

we get

(A.41 Taking

limits in equation

(A.4) as s + 0, and using

(A.3 we get equation (A.1). as was our objective. Since (p(O) = 1,‘5, as shown in Appendix E, equation

APPENDIX

(2.12) follows immediately

B: NON-CONTINUABILITY

We show here that the DE relaxation Indeed, since G(s) is proportional to

modulus,

OF

G(s), has a natural

THE

from equations

MEMORY

boundary

(2.9) and (A.l).

FUNCTION

in the complex s-plane at m(s) = 0. (B.1)

,zd e-pj’:T* we can achieve

our end by setting

z = exp( - s/T,) and showing t 1.1

has a natural A classical

that the function

z,2t-ll’

of z defined for 1~1< 1 by (B.2)

boundary at its circle of convergence, IL( = 1. result, known as a gap theorem [16]. is that if a series

E ad’.

t-1

4 f

0,

k = 1,2,3,

..

(8.3)

Dynamic viscosity of the Doi-Edwards

305

fluid

satisfies k

lim - = 0 k-= ok

(B.4)

then the function defined by the series has no analytic continuation beyond its circle of convergence. For the series (B.2). then, k = 1,2,3, . . . ax= 1, n, = (2k - l)z, (B.9 and equation (B.4) holds, whence the function defined by equation (B.2) cannot be continued beyond 1~1= 1. Since 131= 1 corresponds to s(s) = 0, we conclude that G(s) has a natural boundary at the imaginary s-axis, as was to be shown.

APPENDIX

C: BOUNDEDNESS

OF cp, (2, AND THEIR

DERIVATIVES

Our purpose here is to show that each of the derivatives of 9(7) and ofd(79)/dy is bounded for all real y and that, indeed, they each approach zero at least as fast as y-” as p * co. To begin with, note that standard algebraic manipufation allows one to combine the two terms in the integrand of equation (2.6) and subsequently cancel 7’ to get dx #X2

- l)Z + 4y2x4[1 +

(Cl)

(,/sxa - l)Z + 4J2X’1

which is analytic in a neighborhood of y = 0, so that our statement about the boundedness of 9 and d(79)/&; holds near 7 = 0, i.e. for some y. > 0. our statement holds for y 2 y,,. To complete our task, then, it will suffice to show that each derivative of y’9fP) is bounded for 7 > yo, Note that equation (2.6) gives (C.2) 2?‘9(‘l) = f + H(Y) where 1 p*x*- 1 H(7) = dr = (C.3) 411:*d y * [(7fxZ - I)* + 4{*x’]“2 .

I

and where the second equality is obtained by making the change of variable of integration y = rt. Clearly. the integrand in the first integral in equation (C.3) is bounded by unity, which, together with equation (C.2). gives (C.4) IH( 5 1. 1729(7)1I, 1. To complete our task, then, we need only show that H’“‘(?) is bounded for 7 > yo, for each n = 1,2, . . _ . We shall show that H’“‘(.I) is the sum of terms of the following types: (a) Functions of 7 involving no integrals and such that these functions and all their derivatives are bounded for y>;,; (b) Terms of the form p(
Cn.

c72(i2

_

_ 1)2

p+ +

1

4;4~(2rn+11i2dT*

m= 1.2 . . . . . n,

k s 2m

(C.5)

where C,, are constants and no summation on indices is implied. It should be clear that since the integrand is .,k-2n- * times a quantity which cannot exceed unity in absolute value. and since k - 2m - 1 $ - 1, the integrals in (B.5) are bounded for 7 > yo. To establish our results, we proceed by mathematical induction, First note that an examination of equation (C.3) will show that our assertion holds for n = 0. Next. suppose that our assertion holds for an n & 0. Then differentiation of each term of the form (C.5) gives

(C.6) The first term is clearly of type (a), and the two integrals are of type (b). This establishes the result.

APPENDIX

D: INTERCHANGEABILITY

OF SUMMATIONS

AND INTEGRATIONS

We shall have to interchange summations and integrations in the expressions involving the DE fluid shear functions. We look at D 9= d.7 9(Ks)e -*p’(s) 0.1) I0 where p’(s) = 1 e-+’ and 191 s &f (D.2) Ecdd for some M > 0. Indeed 9 = R_T lim e’ rzd 9(Ks)e-‘rz+‘ouds. 1 r-o

(D.3)

Now, since the series is uniformly convergent for E < s, we may interchange the order of summation and

306

B. BERNSTEIK and R. R.

HWLGOL

integration and then obtain (D.4) Thus convergence of the series of integrals is uniform for all Eand R, whence we get that the limit in equation (D.2) is the sum of the limits of the integrals, and thus, since this argument can be applied to the derivatives of 9 also, we obtain ** cp’“‘(Ks)e-fin’(s)d = 1 z cp’“‘(Ks)e-‘p*‘““ds (D.5) J0 po4dd I0 forn = 0, 1.2,. . . . We obtain a similar result if o, is replaced by 15.These considerations justify the interchanges of the integrations with the summations.

APPENDIX

E: EVALIJATIOK

OF DERIVATIVES

OF $7(y) AT 7 = 0

By direct substitution of 2 = 0 into equation (Cl), one obtains tp(0) = 4.

(E.I)

The calculation of the derivatives of p at ; = 0 could become quite tedious. For this reason, we base developed an iterative scheme to evaluate, as functions of x, the derivatives with respect to y of the integrand of equation (2.6) at 7 = 0. We shall see that these integrands then become polynomials in x, and so the integrals are readily evaluated. Note that since rp(y)is an even function of 7. all odd-order derivatives vanish at 7 = 0. If we put i. = 7”. we get. by comparing the coefficients of the Taylor series expansion of u, in powers of y with the expansion in powers of 1

=(c$dg I

?- 0

We define y and r and $ by yL

!

63)

i-0

- (1 - i..v’) f $( I - i.x’)’ + 4i.x“ (E.3)

2i. : = \.‘(I - i.x2)z + 42.x’

(E.4)

so that $ is the integrand of equation (2.6) (after l;-,‘7.~ has been absorbed within the integral sign). Observing that equation (E.3) has the form of the solution to a quadratic equation, we obtain with the use of equations (E.4) and (E.5) i}l t (1 - i..r”)y - .t? = 0

(E.6)

z = 2i.y + I - ix2

(E.7)

ill: = y.

(E.8)

)‘ = i.(.v’y - y2) -t x4.

(E.9)

Now, (E.6) gives

Successive differentiation with respect to i. gives ,n, )’ = &y

- yZ)L”- 1) + ;.(+

- #‘.

(E.lO)

At i. = 0, equations (E.9) and (E.10) yield y 5z.z ,x4,

y’“’ ‘1 = (n + l)(x2y - y2)‘“’

(E.11)

= xry’“’

(E.12)

and an inductive argument will give, at i = 0. (_u2y_ y’)‘“’

_

whence, with equation (E.11). one obtains (E.13) Successive differentiation of equation (E.7) gives, at i = 0, r=l ,” = 2y - .$

(E.14)

.I”, = Znyl”- II n=2.3,... and equation (E.8) yields (E.15)

Dynamic The net

result

recursive

iS a Set Of

viscosity

relations

of the Dot-Edwards

for calculating

fluid

307

ij’“‘(.x) at i = 0, namely

the following:

): = x1 (E.16)

Thus. for example,

, - ).z = ,p - g y’ = .x-L’ y” = 2(x5

- 2~‘)“) = 2x8 - 6x’O + 4x”

*’ = L” + x2.x* - 2yij = 2x6 - 318

= 2.x8 - 6xi” + 4.~‘~ + 2.+x6

- 3.9) - 4xJ(lx6

- 3x3) - 4(x6 - x8).x

= 2x8 - 24.x’O + 20.r’z SO that the derivatives

APPENDIX Here we obtain

of cp with respect to 7 at ;’ = 0 are given by

F:

LIMITING

the expressions

EXPRESSIONS

AT INFIIiITE

FREQUENCY

for

(F.1) Indeed,

write P.2

E, =

I

e,(x) dx.

e&x) =

P

I 1 2 ( pz + iR)‘”

I l

’ - (.x2 + iR)*”

’1’ p= 1.3.5

l

)

(F.2)

We have 1 (p’ + ifl)‘“’

1

I

1= ” + z

(,~L+ iR)2”+I d.~

(F.3)

and

Our first task is to show that the sum on the right-hand we shall evaluate the corresponding limit of the integral We note that 1 (p2 + iR)1”‘1 -(x2

side of equation (F.4) approaches zero as R + x. Then on the right-hand side of equation (F.4).

(,yz + fi)2” + ’ _ ( pz + iQ)2”+ ’ 1 + iR)Z”‘1 = (p’ + iR)Z”‘1(,~* + i*)*“+l = (I~ _ p2)(x2 + ill)‘” + (x2 + ifI)‘“-‘(p’ (p* + iR)Z”“(x’

= (x2 - p’)

+ .‘. and therefore

+

. . + (p2 + in)‘*

+ iR)*“+’

1

1

(p2 + iR)*“+’ (x2 + iR) + (p’ + iR)z”(x’ + iR)x 1

(p* + ifl)(x* + iR)‘” +

since we consider p 5 x 5 p + 2 in equation Z(P +

,eP, * (2n + 1)(.x2 - p’) Z(pS + fly-

+ in) +

s

’1

F.5)

(F.2).

1)(2n + 1)

(p’ + fPy+l

4p(2n + 1) d (p’ + fly+

(F.6)

and l-V&,1

s

8pJ?i(?n

+ 1)

pa + I21

’ ~ 8pfi(Zn

+ 1)

p4 + f-2’

(F.7)

308 Since for R z 0, IF.8)

we get (F.9) This is sufficient to show that the series on the right-hand side of equarlon (F.4) converges uniformly in R, whence the limit of the sum is the sum of the limits. the latter being zero as R - x. Therefore the limit that we seek is that of the integral on the right-hand side of equation (F.4). A change of variable of integration

to I = .~!,.‘z

gives

1

(F.10)

The

theory

V ‘z

= (-

of residues

I + i),‘,:5,

allows

us to evaluate

this integral

by

finding

the

residue

of the

integrand

at

namely we have

(F.1 I) Furthermore,

since

(Ks)~c .M for 1c+dznr2’

some .\I > 0. we get for the integral terms in equation

(4.6)

(F.12) whence the absolute value of the sum of the integral

terms in equdlion

(1.6) is bounded by (F.13)

and (F.14)

so that cp”“+~‘(Ks)e-‘P”i”“d.~ which will be required to establish the validity of the asymptotic

series

= 0

(F.15)