180 A TURIN [MarcG . M. h 16,...182 A TURING [Marc. 16, M. h a similar integral ove — 1r / 0 by Oi{u). Then by a, change of variables, we obtain and, by the theorem of residues,

180 A. M. TURING [March 16,

A METHOD FOE THE CALCULATION OF THE ZETA-FUNCTION

By A. M. TUBING.

[Received 7 March, 1939.—Read 16 March, 1939.]

An asymptotic series for the zeta-function was found by Riemannand has been published by Siegel*, and applied by Titchmarsh f to thecalculation of the approximate positions of some of the zeros of thefunction. It is difficult to obtain satisfactory estimates for the remainderswith this asymptotic series, as may be seen from the first of these twopapers of Titchmarsh, unless t is very large. In the present paper amethod of calculation will be described, which, like the asymptoticformula, is based on the approximate functional equation; it is applicablefor all values of 5. It is likely to be most valuable for a range of t where/ is neither so small that the Euler-Maclaurin summation method can beused {e.g. t > 30) nor large enough for the Riemann-Siegel asymptoticformula (e.g. t < 1000).

Roughly speaking, the method is to use the approximate functionalequation for the zeta-function, with the remainder expressed as an integral,

f*which for the moment we write as h(x)dz. We approximate to theJ-QO

integral by the obvious sum 2 — hi — I and we find that, if certain modi-fications are made in the "main series", this gives a remarkably accurateresult; when the number of terms taken is T=2K-\-l the error is ofthe order of magnitude of e~*nT. The theta-functions give another caseof this phenomenon. We have the identity

00 1 00

2 e-™*x=-L- 2 e-m2/x

•yx n=_oon = —oo

* (J. L. iSiegel, "Uber Riemanns Nachlass zur analytischen Zahlentheorie", Quell.Gesck. Math., B, 2 (1931), 45-80.

| E. C. Titchmarsh, " The zeros of the Riemann zeta-function ", Proc. Royal Soc. (A),ir.I (19:35), i':{4-255; also 157 (1936), 261-263.

1939.] A METHOD FOR THE CALCULATION OF THE ZETA-FUNCTION. 181

for any positive x. Hence

1 £ e - ^ ! (if i c ^ 1),

K /cr r f S /«8

As we have proved it above, this inequality depends entirely on thespecial form of the function e'™2. But we can also prove it in this way.We integrate the function e-*M7(l—e~2iriM(C) round the rectangle with thevertices ±i2±iK. In the limit 72->oo we obtain, by the theorem ofresidues,

f—oo+i«/ fe(

».e.

fOO-lIC fCO-t<C p-WUZ p-ZwlUK f—CO+IK p-WU2

p2wlUK«o 1 e I2,iu. du+\— oo— tie A c J oo+t/c

The path of integration on the left-hand side of the equation can bereplaced by the real axis, while the right-hand side is less in modulus than

2 f00Zjj t lexP t~~7T(U~iK)z~2rrt(w— ii<)K)\du

1 6 J—oo

This argument can be used in more general cases, but in moving the pathof integration we may encounter singularities of the integrand, whichwill modify the result.

We base our calculation on an integral representation of the zeta-function due to Riemann*.

1. Evaluation of a definite integral. Let

_ [ exp (iTTZ2+Ziriuz) dzettrZ o—inZ '

where OS 1 signifies that the integration is along a line from — eoo toeoo cutting the real axis between 0 and 1, and e = eiwi. We denote

* Siegel, loc, cit., 24.


a similar integral over — 1 / 0 by Oi{u). Then, by a change of variables,we obtain

and, by the theorem of residues,

O(u) = 0,(^

Multiplying the identity

by exp(iVz2+27rtWz), and integrating over 0 / 1, we obtain

Combining these results we have

2. An integral representation of the zeta-function. We integratez~al(l—e~27rie) round a curve L which may be taken to consist of a straightline from ioo to h, a semicircle from £ to — \ lying in the lower half-plane,and a straight line from — \ to ioo. We define zr8 so that it has its usualvalue on the positive real axis, and is continuous except on the positiveimaginary axis. The integral round any part of a circle with 0 as centretends to 0 as the radius tends to infinity through appropriate values,provided that &$ > 1, and therefore in this case we have

f z~s dz z~8

z s-j; = 27rt (sum of residues of •= srji at integers other than 0)jjj i—e i—e

00V^ / Q I fifO Q\ y

This gives the analytic continuation of t,{s) over the whole plane exceptpossibly for even integers. Now, by (1.1),

u~8

L

u-se%«iuzdu]dz

fo/ I

1939 . ] A METHOD FOR THE CALCULATION OF THE ZETA-FUNCTION. 183

i.e.

£(5)(l+e-s) = f 6J^~1^-J L ^ e

I n t h e first i n t e g r a l w e m a y r e p l a c e L b y t h e t w o l ines O i l a n d — 1 1 0,a n d w e g e t

er-tû-'duf e-inu*u-*du _ f

JL elVu-e-|VM ~Joxi

w-e- 'V u e - 'V u-e l V u/

einu_e-irru

The curve may now be replaced by 0 \ 1 if the sign is changed, and wethus obtain

The zeta-function is then expressed in the form

TSr^-SW1 BinWrd-.) ^ z l l (2.1)

and the calculation of £(s) is reduced to that of the integral

* w * : ( 2 - 2 )

forlOM

If we multiply both sides of (2.1) by T(^8)7ris, and make use of therelation

we obtain

» i ^ «), ( 2 . 3 )

and on the critical line this is equal to

(2.4)

For points on this line there is, therefore, only one real integral to calculate.For points not on the line there are four real integrals.

3. The method of calculation. Let /u be a positive real number lyingbetween the integers m and w + 1 . Then

/(a) = [ h(z)dz = \ h(z)dz~ 2 r~«.JO/l Jm/m+l r=l


Now let K be a positive real number and put

, w = m1—exp [—2TTK€{Z—ft)]'

The function g has simple poles at the integer points other than 0. and atthe points pk = (i-\-ekJK, where k is an integer; otherwise it is regularexcept on the closed positive imaginary axis. The residue at the non-zerointeger r is

^2m{\— exp [— 2TTK€{r—/x)]}'

and at pk it is ^ k ' .• K 2TTKC

The line on which the poles pk lie, taken as running from left to right,will be called P.

Let J and J' be two curves going from the third quadrant to the firstand from the first to the third respectively, J being entirely on the right ofP and J' entirely on its left and on the right of the origin. Suppose alsothat there is a positive real number a such that, at sufficiently greatdistance from the origin, the curves lie in the region where eithera < argz < %7r—a or \-n—a > argz > — Tr+a, and that the length of curvewith \z\ < R is 0{R). Then it is easily seen that

g(z)dz = 2iri (sum of residues of g between J and J')J+J'

T~s

where the first sum is taken over the integers r lying between J and J'.Now

= h(z) exp [—2TTK<-(Z—a)] j [ 7/ v 7-z Z-t-f— v r J.-, dz+ h(z)dz,l-exp[-2wce(2-/a)] ^ J P

and g(z)dz — — h(z) -— r<> / \-i dz.)j'JK ' }j> l-exp[27r/ce(2-^)]

If the curves J and J' are always distant more than J/c"1 from the line P,then we have on them

1—exp [2m<€8g(z){z—fi)]1-27.

1939 . ] A METHOD FOB THE CALCULATION OF THE ZETA-FUNCTION. 185

where ag(z) has the value 1 or — 1 according as z is to the left or to theright of P. We can now collect our results in the form

0k K r=i

where 8r has the value 1 if r is on the left of «/', the value

{1—exp [2iTK€(r—/M)]}-1

if r is between J and J', and the value 0 otherwise. The remainder RQ

satisfies

\ 1-27 exp [ - V2HK/* sg(z)] [ e J ^ t > g | e**M|dz|, (3 . 2)

where <£(z) = i7rz2+2-mcez sg(2) — i£ logz. (3 . 3)

We may also write the formula for I(s) in the form

* 6 eiwpi2pTa °° r~B

t= co T e ' ^ - e - ^ , ? ! 1-exp

where B=R0-{-R1, pk = (i+ek/K and

I ^ I < 1 -27Sr~8 exp [— \/2TTK\r—fj.j],

the summation being over positive integers not between J and J'. Byexpressing /(s) in this form we can eliminate from the numerical calculationany reference to the position of the curves J, J', and the remainder isnot appreciably increased. In § 5 we choose the curves so that R1 = 0.Of course, in the calculation the factor {1—exp [27r/ce(r—/x)]}-1 willbe put equal to 1 or to 0 except for a comparatively small number ofterms.

In estimating the remainder we suppose that a > 0, but this is notnecessary.

4. General remarks on the estimation of the remainder. Suppose that

Then, for any curve C deformable into Co in the domain of regularity of

(4.1)

Now suppose that &w(z) has large and fast variations, while \k(z)\ iscomparatively steady. Then the value of the integral (4.1) is


principally affected by the maximum value of H3Uy(z) on C, and a goodinequality for \U\ is obtained by minimising this maximum. It iseasily seen that if there is a curve for which the maximum is minimised,and if z0 is a point at which the maximum on this curve is attained, thenw'(zQ) = 0, i.e. z0 is a "saddle point" of w. Suppose that in the neigh-bourhood of the saddle point the curve is z = zo-\-le

ia, I being an arc-lengthparameter. Then the contribution to the integral from the neighbour-hood of the saddle point is approximately

We naturally choose a to be £77— £ arg«/'(?0), and then the expressionbecomes

In the estimation of Ro we could take w(z) to be either <j>(z) or<j>{z)±iTTZ, in the latter case different signs being taken on the two curvesJ, J'. With the first of these forms the analysis is simpler, but thesecond gives a better result; we deal only with the simpler form.

We actually use the idea of saddle-point integration in the followingform. Suppose that we have a curve C with arc-length parameter Ibeginning at 1 = 0. Then

dz(a) if H^'(z) JT ^ — al on the curve, where a > 0, then

on the curve, and consequently

This enables us to estimate the contribution to the integral from theneighbourhood of the saddle-point; we may estimate the contributionfrom the remainder of the curve by one or more applications of

(6) if »^'(z) ^y <, - a < 0 on the curve, then e » W " < e - ^ W W and

a

5. Detailed estimation of the remainder. For the two curves J, J' wehave two different saddle-points z0, z0', these being zeros of <j>'{z) on the

1 9 3 9 . ] A METHOD FOR THE CALCULATION OF THE ZETA-FUNOTION. 187

two sides of the line P . We may put

it/z = — (z—zQ){z—zx)

on the left of P, and

. 27rt ,. ,.

Z

on the right of P; the points z0, z0' are in the right half-plane and zx, zx'in the left. If we put further T = t/2n, p = /CT~}, t, = ZT"*, we may writethese equations as

The roots satisfy

Co+k'Ô, d + C o ^ O , arg£o+arg£o' = O, | w

There is a cubic curve independent of p on which all four roots lie. Weshall need a number of other properties of the roots, and we mentionthem as we require them; they are mostly inequalities which can beproved by straightforward but laborious methods.

The behaviour of a number of functions of p is shown in Fig. 1.

05

188 A. M. TUBING [March 16,

We choose the curves of integration J, J' as follows. J consists ofthree straight parts Jv J2> J3, of which Jy is a straight line from — eoothrough z0 to 6, where b = zo—lyo{l+i); J2 joins 6 to 6+jS, where j8 isreal and positive; and J3 joins 6+jS to eoo, passing through a hah0 oddinteger. As j3 tends to infinity the contribution to the remainder fromJ3 tends to zero and the contribution from J2 tends to a limit. We maytherefore omit J 3 and suppose J to consist of J1 together with J2, takenas extending to infinity; there are then no poles on the right of J tocontribute to Rv

Fig. 2.

The curve J' consists of the four parts J^, J2', Js', J I, of which J2'is further divided into J5' and «76\ Let 6' = z0'—%yQ' (1+i). Then J /is a straight line from e oo to b' through z0'; J2 is a straight line from b'to c, where c/b' is real, 0 < c/b' < 1, and | c | = min (£, ^| z0' |). The curveJ3' is part of a circle lying in the half-plane %zz > 0 and joining c to — ic.We obtain J4 ' by reflecting Jx' and J2 in the line 3ez = 0 and reversingits direction. We divide J2 into Jg' and J6

r, of which J5' is the part forwhich y > §yQ'. Both of these parts are of positive length, J6 ' being oflength £ | ZQ I at least.

1939 . ] A METHOD FOR THE CALCULATION OF THE ZETA-FTJNCTION. 189

andOn Jx and J2 we have \z\

1£ cosecg—i

The contribution to Ro from J is therefore at most

i(l-27)i-*»(»rf0)-' exp cosech o| [ z\. (5.1)

On Jx we have 0 > arg (z—zj/z > — cos"1 J, a simple consequence of thefacts that the distance of Jx from 0 is greater than 2~41 zx | and thatarg zx < — §TT. Also

| 2n(z-Z1)(Z-ZQ)/Z I > 7T | Z-Zo |,

so that.. 2iri(z—z-.)(z—zn) dz , , , i ,£ * f± &• M < -$TTZ = -4TT| 2 -2O| ,

if I always decreases as z0 is approached. Then, applying (a) of § 4, we have

f e»*fr>|<fe|<-v/6e»«*o>. (5.2)

Also ^<j>{b)<^4{zQ)-\7ry^. (5.3)

On J

But T < a;02, since | 0 > 1 for all p, and therefore

(5 . 4)

Consequently, by (6) of §4, (5.3) and (5.4),

f (5.5)

Now let us turn to J'. If z is a point of «/1/+«/2> ^ n e n ~*^ ^

corresponding point of e/4', and

-iz) = $i[>7r(—iz :(—iz)—it logz—it log (—iz/z)]

Hz—it log z—££7r],

i.e.

Also | —iz | = | z | and

cosec (— ivz) | < | cosec TTZ j ,


as may be proved by the use of the product representation of the sinefunction, remembering that jargz|<£7r. Consequently

On J-l we have 0 >arg (z—22')/z > y—îr, where y = \TT—arg£0', and

|27r(z-z1')(z-z0/)/z| > 27r|z-zo'|.

Therefore »<£' (z) -jj < — 2TT sin y

on J-l, if Z decreases as z0' is approached from either side. Applying (a)of § 4, we have

( W W l K ^ (5.7)

and 3Ro(6) < &<f>(zQ )—^ny'^2 s m 7 - (5. 8)

On Jt'

„ ,,, dz . . — 2TTIZ

y—Zo')(b'—z1t). (5.9)

But arg {z0'—b') = |TT and

0 > a.rg (6'—zx') > arg ( — zx') = y —^w,

so that

^7r+y < arg -rrrr (6'— •Zo')(b'—Zi) < hT-

Also |-27n(6'-zo ')(6'-2 l ')/!6' | |>V27rt/o ' ,

so that

& | »/i (6'—zo')(6'—z/) < — \/27rt/o' smVj (5-10)

and applying (6) of §4, and using (5.9), (5.10) and (5.8), we obtain

j e*«*>|(fe|< (V2ni/o/siny)-1exp[3a0(2o')-^'rt/o28my]- (5-H)

1939 . ] A METHOD FOR THE OALCTJLATION OF THE ZETA-FUNCTION. 191

Also

<

On J3 '

arg

so that on both «/6' and J3 '6 and J 3

zo'|siny. (5.12)

< 2 sin"1 £ < \n < £TT—y;

siny.

This gives, using also (5.12),

< (y27rt/0' siny)-1 exp [itffo'HHo* siny—JV2ô'l2o'I s inrl• (5 •13)On J{ we have

; ir-i<7) ££0' h* cosech j \ny§ I; (5.15)

on

0' |-a cosech j ^Trt/o' | ; (5.16)

and on «/8', Js'

[min (fc ~l coeec arg^0' . (5.17)

We may now collect our results to give an inequality for \R\. Weuse (5.1), (5.2), (5.5), (5.6), (5.7), (5.11), (5.13), (5.15), (5.16) and(5.17), and we make the exponents ^(zo)+v/2Tr/c/u, and &<f>(zo

f)—more explicit by the use of the relation z0

2 = i/czo+r:

\E\ = | Ro\ < 0.635

X exp [ -

^** cosech j

arg

2/o

X j (sin y)-*- 2ff cosech | 7rt/0' | + (V 2 ^ o ' s i n r)"1 e xP [ " W s i n

'l + IS'l'^C (iX [ 3 ' cosechL

o'!)]"'"1 cosec arg^0'

exp [ - '] . (5.18)


At this point it may be as well to repeat the definitions of the variousquantities appearing on the right-hand side of this inequality, in termsof s, K:

The complex numbers £0', £•/, £1} £0 are the roots of the equation

( £ 2 - l ) 2 = -ipH*

lying respectively in the first, second, third and fourth quadrants, and

where £, 77, x, y are real.The estimate (5.18) for R is somewhat complicated. It may be

simplified considerably when p is not large. I give an estimate for thecase p \ ; we then have

siny>0-55, (5.19)

The result is that, for p \} t > 25,

| R\ = | RQ\ < r-*» fo-76 . 2i-^ cosechO-78(/c'/V2)

X •;2-45+0-40(K/V2)-1exp(-0-13(/</V2)24e

X I 1-91.2"-i cosechO-64(/c/-v/2)+l-00(/c/V2)-1exp ( - 0 ' 1 4 ( K / V 2 ) 2 )

X jr74.

X (K/V2) exp ( -O-ISTK/C/V 2 ) )} ] ^~B\ (5. 20)

where

A = 77x&e(z0— 2/i,) + 27rr arg^0', B = 7r/c1&e(2/x—z0')—2TTT arg£0'.

In the case p \ the estimation of the remainder from (5.18) may bemade easier by the use of the following inequalities, which are valid forall positive p:

l ^f| ^ . (5.21)


6. Choice of parameters. The remainder xoith a finite, series. Theremainder R as given by (5.18) is the sum of two terms in which the mainfactors are e~A and e~B. The most favourable choice of the parameter fxis presumably approximately that which makes these two factors equal.Calling this value /x0, we have

~J a r g Jo'J

As p tends to infinity, /x0'—'/c/2v"2; and, as p tends to 0, [XQ~TK Also

£(Zo-3zo') + y arg V

As p tends to 0, the factor

tends to 1; and, as p tends to infinity, this factor tends to ^. For all positivevalues of p it is greater than £, and therefore

Also Ke(2o-/-*o) = i [ l ^ ( 3 2 0 - V ) - y arg ^ j ,

and we have 5 e(z0—JU,0) > \K.

If for /n we choose 7xo±3, wliere 8 > 0, then the greater of the exponentialfactors e~A, e~B is

exp[—£7r

The values of /x which we may choose are restricted only by the conditionthat the curves J. J' must be distant at least 1/(4K) from P. If K -\/2,we may then choose p in the interval (/x0, ^ 0 + | ) ; and, if K 2, we maychoose it in the interval (/x0—£, /xo+|). However, for such small valuesof /c it will probably be best to choose /x rather close to /x0. We neednot consider the case of smaller values of K than -\/2, since, as will appear,

9B». 2. vol.. 48. NO. 2331, 0


it is not advantageous to take such small values, even when there is onlyone term taken from the series lth(pk).

When p is small, /x0 is close to T*, and it is therefore probablysimplest to choose a value of fi which is close to T* without actuallycalculating /x0. The inequality

should then be of value.For large values of K (e.g. K > 3) we shall do well to choose fi to be

either an integer or a half odd integer. In the case that /x is an integerthe function g has a double pole at /x; in place of the terms

e r~8

7 h{fl)~ l-exp[2nire0i-/*)]

we have therefore to put the residue at 0 of

2-rnj — Y (z+/*)~s exp [^(z-f/x)2—iirz]

and this is equal to

In the practical applications of course we take only a finitenumber of terms from the series S (e//c)A(jpfc). We therefore wantan estimate of the error arising from this. If we decide what is thegreatest total error r) we can admit in our result we can proceed in thisway. We choose K SO that \R\<fyr) and then take sufficiently manyterms of the series for the error from this second source not to exceed £17.Now let us estimate this second remainder. For this purpose we prove thefollowing

LEMMA. The function \eiwziz~u\ has only one maximum on the line P.

Put z = /x(1+0(1 +i)), a = */(2TT/X2) ; then 0 is real and

Let us abbreviate the right-hand side to #(0). Then

1939.] A METHOD FOR THE CALCULATION OP THE ZETA-FUNCTION. 195

Now

and therefore ((l+#)2+02)(l+20)—a cannot vanish for more than onevalue of 6', it clearly vanishes for at least one value. Then H{6) has justone stationary value, which is easily seen to be a maximum. This com-pletes the proof of the lemma. If a < 1 the value 6 giving the maximumsatisfies 0 > 6 > a1— 1.

Put uk = K-1 | h(pk) |; then, if a > 0,

If we suppose that ^{p^z-i) *s le s s than the value of y for which themaximum of \einztzru\ occurs, and if k^K+1, then

and therefore

v ^(V-\~a exP[—7r(-^+1)/ 'c\/2+3Sl(*^-ir-i)ll^-i-i |2^ qj^ -** — I —-— — — — — _\y/V ( l - e x p [-^2IT{K+1)IK] ) (1-exp [-TT/K^/2])

Similarly, if (as is always the case if K' ^ 0 and a ^ 1) 3Hpjr+:i) ^8 greaterthan the value of y for which the maximum occurs,

K'+i ( l — o x p [ — V 2 7 r ( - ^ ' + 1 ) / * ] ) (1—exp[—TT/ICV2])

so that

< 2^|A(pA1)| + lA(py+1)!K (1 - e x p [ - y%r(Z*+1)/K] )2 (1 - e x p [-TT/K y/2])'

where Z * = min (IT, Z').In the case in which K is small compared with T* we can easily obtain

a rough estimate of the number of terms required for a given accuracy.For in this case K, K' are such that JO.Z/A1"1 an(^ PK'Ip—l a r e small,

02


and uK is approximately exp [—2TT(K-\-1)2JK2]. If the remainders R andR* are of the same order of magnitude, then we have approximately2<n(K+l)2/K

2 = \TTK2, i.e., K+l = \K*. The number of terms T that wetake is 2K-\-l, i.e., approximately K2, and the total error is of the order ofmagnitude of e~inT. If this statement is to be put into an exact form wemust say that if fj, and K are suitably chosen as functions of t, 77, and alies in the interval 0 ^ a ^ 1, and if rj tends to 0 and t to infinity in sucha way that K'1, KT~* also tend to 0, then the error does not exceed rj andT'1 log??-1 tends to \TT\ this does not hold for any number larger than \TT.

When KT-* is of the order of magnitude of 1 we cannot get so simplean estimate of the number of terms needed, but we can obtain an estimatein the limiting case KT~ 4->OO. We may then neglect all the factors in uh

except |e'*#fe2|. Putting /x+ev = p_K, we have approximately

if R and R* are of the same order of magnitude; i.e.,

But for large values of \i we have approximately /x = K/2^2, and thereforeapproximately

2V2+KV—K2 = 0.

The two roots of this equation are e(P_£—/x) and ~£(PK>—p) (approxi-mately) ; the difference of the roots is f K and therefore T = |K2 approxi-mately, and the error is of the order of magnitude of e~iirT.

It is possible that this might be improved by taking /u, to be somethingdifferent from /x0; for if we take /u, closer to T* the remainder R* is madesmaller for any given value of T. Such an improvement would necessarilybe at the expense of the remainder R; I do not think that any appreciableimprovement really can be made along these lines.

In the case in which T = 3 we may put K ~ 16 -y/2, and then, if a — \,/x = Ti and t> 350, and if the factor 11—exp [2-7r/ce(r—fi)] j " 1 is replacedby 0 or 1 except for two terms of the main series, the error from all sourcesdoes not exceed 0i0044r"i.

7. A similar method. There is an alternative, and better known,integral representation of the /eta-function on which we may base ourcalculations, viz.,

m' l f o2m'i

r=\ r = l 1~f~e JQ


Here Qm is a curve coming from infinity in the first quadrant, crossingthe real axis between m and m+1 and again between — m and —m—1,and going on to infinity in the second quadrant; z~s is defined as in §2.If we choose m = m' = [r*], and let the part of Qm in the neighbourhoodof the positive real axis be a straight line cutting the negatively directedreal axis at /x and at an angle of + JTT, then, for large t, the only appreciablecontribution to the integral comes from the neighbourhood of the positivereal axis. We can approximate to this integral in the same way as before,the resulting approximate value for £(s) being

00

S r~s 11 - (1+efir8)~1 (1—exp [-2TTK€(r—/A)] ) - 1

m

-(8) 2 rs

expa) fc=_£ 1-exp [—

The integer K must not be chosen too large; K < 4r* K is usuallysufficiently small. This method has the advantage that for points not onthe critical line only two real integrals have to be evaluated, and not four.This may be of value for calculation of zeros not on the critical line. Forthis purpose it will not matter that the method is only applicable forlarge values of t; it is, however, possible to remove this restriction byintegrating along a parabola, e.g., the parabola

The conformal mapping u2 = z transforms this parabola into a straightline, so that

f eivimzzr8dz_ f

Imi/ (tn+l)4 -1 c

The Une of integration cuts the imaginary axis between - - imS and — i (m -j-1) *.

King's College,Cambridge.

Documents

180 A TURIN [MarcG . M. h 16,...182 A TURING [Marc. 16, M. h a similar integral ove — 1r / 0 by Oi{u). Then by a, change of variables, we obtain and, by the theorem of residues,