On the Generalization and Application of the Eular Maclaurin Formula

AN ABSTRACT OF THE THESIS OF

Samuel Codjoe Arthur for the M. S. in Mathematics (Name) (Degree) (Major)

...... , a / / _ Date thesis is presented //rE`- C'.`UIG¡l `ll/// 1/ Title ON THE GENERALIZATION AND APPLICATION OF THE EULER - MACLAURIN FORMULA Abstract approved (Major pro essor)

The Euler- MacLaurin sum formula has appeared in the titles of two quite recent papers whose authors were primarily interested in certain applications. In this paper a somewhat different approach to the myriad of formulas for summation, integration, differentiation, etc. , is based on the simple identity which defines the set of

Bernoulli numbers. Variations of this identity are obtained by the most elementary manipulations, then application of the Laplace transformation leads to the well -known formulas, trapezoidal rule,

Simpson's rule, etc. , complete with an infinite series of higher derivatives. This type of formula is particularly valuable in carry- ing out a Frank type of inversion of a Laplace transform. In particu- lar, the Frank method has been extended -to the alternating series case. The representation of error of the approximation formula by means of an integral involving a periodic polynomial has been extended to Simpson's rule, with indication of a general method for extending the theory for more general approximation formulas. ON THE GENERALIZATION AND APPLICATION OF THE EULER MACLAURIN FORMULA

SAMUEL CODJOE ARTHUR

A THESIS

submitted to

OREGON STATE UNIVERSITY

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

June 1964 APPROVED:

Professor of Mathematics

Head of Department Mathematics

Dean of Graduate School

Date thesis is presented i c .r° 4- /' /"Z" Typed by Muriel Davis TABLE OF CONTENTS Page

INTRODUCTION 1

BERNOULLI IDENTITIES 3

EULER -MACLAURIN FORMULAS 14

INTEGRATION PROCEDURES 24 INTRODUCTION

Frank [ 6, p. 89 -91] and Gould and Squire [ 7, p. 44 -52] have made some recent extensions and applications of the well -known

Euler -MacLaurin sum formula. Authors pf many textbooks and treatises have employed this method to arrive at estimates of such finite sums as N

1 / n = log N Y - 1/2N - 1/12N2+ 1/120N4 -.. U n=1 or of the - factorial

n! = nn en exp[a/12(n-1)], 0< a< 1. Frank showed that the geometric series expansion of certain types of Laplace transforms may be summed in such a way that the inversion yields a rapidly converging series regardless of whether or not the time variable is taken to be large or small. This method re- quires the infinite series form of the Euler- MacLaurin formula, rather than a finite series. All integration and differentiation formulas obtained below will have this feature, something not usually found in the standard texts. Gould and Squire developed a second form of the Euler -MacLaurin formula with. the important property that the algebraic sign of the estimated remainder term would be opposite that of the corresponding term for the first form. Their 2 formula has been called the second Euler -MacLaurin formula by

Hildebrand [ 8, p. 154] .

The purpose of this paper is to (a) develop a number of these types of formulas by elementary operations, (b) extend the Gould and

Squire and Frank results to the alternating series case, and (c) study the nature of the error associated with the several approximation formulas. 3

BERNOULLI IDENTITIES

The basic identity is No. (1) of Erdelyi, et al. [ 4, p. 51]

2k-1 1 +e -z B 2k (1) < 2 Tr, e-z (2k)!z Izl 1 - e k-o where the B2k are the Bernoulli numbers. A partial list is

1 1 1 1 5 691 Bo 1, B2 = ' B 4 = 30 ' B6 = 42 ' B8 = 30' B10 66' B12- 2730 '

7 3617 43867 174611 854513 B14 6' B16 510 ' B18 798 ' B20 330 ' B22 - 138 '

236364091 8553103 23749461029 8615841276005 B24 2730 ' B26 6 ' B28 870 ""B30 14322 '

. . A list complete through B60 is given by Fort [5, p. 49] ;.prob- ably the most extensive list was published by Adams [ 1,p. 259 -272], all the way through B124. It is interesting to note that B120 has

2358255930 for a denominator while B122. has 6 for a denominator and a number with 107 digits for numerator. B2k It is preferable to C2k =_ k = 0, 1, 2, , set 2k , ... since these are the coefficients of interest in the several developments below:

1 1 1 1 1 CO = 1, C2_ C4_ - 12' 720' C6-30240' C 1209600' C10 47900160

C12 = -5. 350748 10 10,..., The basic identity may be written as 4

co 1+e-z- 2k-1 (2) 2 C z Iz I < arr . - e-z 2k l e =o

The restriction on the value of I z need not be repeated in the development of additional identities. It is required since the B2k grow very rapidly; an asymptotic form is

B2k 2(-1)k-1 C 2k (2k)! (2) 2k given by Knopp [ 9, p. 527 ] .

Additional forms of the basic identity depend on the identity

(1- z)(l +z+... +zn -1) = 1 -zn. The first such extended form is

(1+e-z)2 1+e-2z z2k-1 (4) - a - [1_22k-1a] 1-e-2z 2k 1-e-2z k=o There are three ideas governing the choice of values of this undetermined parameter, and of others introduced below, the development of (1) an open end integration formula, (2) a closed end integration formula, and (3) a differentiation formula. In each case the complete infinite series expansion is included. Equation (2), as it stands, will lead to the simplest closed end integration formula, the trapezoidal rule. Equation (4) cannot lead to an open end formula but it does lead at once to the well -known Simpson rule and variations of same if it be required that

2k-1 (5) 1 - 2 a = 0, k = 1,2,3,... 5

It is convenient to consider the first three values of k; for a = 2

Eq. (4) becomes

Zz co 1 1 + 4e-z e 1 4 2k-2 2k-1 (6) + -1)z 3 1-e -2z - z 3 C2k(2 k=2 For a =8,

1 7+ 16e z+7e 2z 1 4 16 2k-4 2k-1 (7) . 15 -2z = z+ C2z 15 C2k(2 -1)z 1-e k=3

1 For a = 32 ,

1 31+ 64e z+ 31e-2z 1 ' 20 16 3 64 ( 2k-6 2k-1 (8) - C2z+ . z+ 21 C2z+ ZiC4Z 63 C2k -1)z 63 1-e-2z k=4 The formula for differentiation results if k is taken to be zero in

Eq. (5), which yields a = 2 and Eq. (4) becomes

1 l (9) -::z2z C2z + 3 Y 6 -2z _ C2k(21)z ' 1 - e k 2

The next step is to write down the identity

2e-Zz+e-3z 1+e-3z ro 1+ 2e-z+ +e [1_32k-lai Z k-1 (10) - a - C -3z 2 1-e 3z 1-e k=o and require that 2k-1 (11) 1 - 3 a = 0, k = 1, 2, 3, . For a = 3,

3 1+ 3e-z+3e 2z+e 3z 1 (12) 9 C2k(3 -1)z z- 2k 8 1-e-3z 8 k=2 6

1 For a = 27 , -2z -3z 00 3 13+27e +27e 3z 1+ 2k-1 (13) 9 C z- 88-1 C(32k 80 -3z z+10 C2z 80 2k 1-e k=3

1 For a = 243 ro -z -2z -3z 1 90 81 3 729 2k-1 121+243 +243e +121e =-+-C 32k-6-1) z (14) 3 z92z+91C4z -728 2k ' 1-e k=4

The formula for differentiation results if k is taken to be zero in

Eq. (11), which yields a = 3 and Eq. (10) becomes

-2z -33z 1 1- e-z-e 1 2k 2k-1 - C2z + C2k(3 -1)z . ( 15) 4 -3z 8 1 - e k=2 The identity,

-z -2z -3z -4z 2 2 -4z (16) 1+2e +2e +2e +e (1+é 1+e -a ) +b 1-e-4z 1-ez 1-é z

1a+24k 2k-1 = 2 C 1-22k 2k b z , k-o leads to open end integration formulas if it is required that

1 -a =0 (17) 2 k-la. 4k-2 1 -2 a + b = 0, k = 1, 2,3... The solution is at once 24k-2 22k-1 24k-2-1 1 ( 1 8 ) a k = 1, 2, 3, . . . 24k-2-22k-1' b - 24k-2-22k-1' 7

For a= 2, , b = , the identity of Eq. (16) becomes

-z -2z -3z ro 4 + 2e 1 (19) 8 [ 1-3. 22k-2 24k-3 z2k-1 - z 3 3 1-e 2

For a =8,9 b =8,

00 4 8ez=e2 +8e3z 1 8 32 4k-5 2k-1 ( 20) C 1-9.2 2k-4+2 15 z 5 C2z+ 15 z k=3

331-e 1 For a = b = 32 , 32 '

4 32e z-e2z+32e3z 1 40 3 (21) -32 C 63 4z z 21 3 1-e C2z oz

co 128 2k-6 4k-7 2k-1 2 +2 ] z . + 63 C2k[ 1-33 k=4

The closed end formulas result if it be required that

22k -1 24k -2b 1 - = 0 (22) 2k+ 1 4k+ 2 1 - 2 a+ 2 b = 0, k - 1,2,3 ... The solution is at once

5 (23) a = , b - , k = 1, 2, 3, . . . 22k+1 24k '

For a = b = , the identity of Eq. (16) becomes 8 16b

-z -2z 2 7+32e +7e4z 2k-4 24k- 2k-1 ( 24) 1 64 [1-5.2 z 45 --+-z 45 C k 1-e k=3 5 For a b - 1 = 32 ' 256 '

217+512e-z 1 (25) 2 +432eZ+512eá +217z 16 945 -4z -21 z + C2z

2k-6 4k-10, 2k-1 1024 ¡¡ +2 z - . +945 ,C21c`1-5.2 k=4

5 1 For a = b - 128 , 4096 '

2 3937+8192e +7872e 2 +8192e 3z+3937e 4z 1 112 256 3 16065 1-e-4z = z 1192z 357C4z

oo (26) + 16384,`C k_1 1-5.22k-8+24k11Z 16065 J k=5 The differentiation formula results if it be required that

-4=0 1 -2+b (27) 22k-1 a 1 - + 24k 2b = 0, k = 2, 3, . . . The solution is at once

2(24k - .1) (28) a = 1) b 4(22k - 24k 22k ' - 24k 22k ' k=2,3,... -

18 For a = , b =- , Eq. (16) takes the form

00 1 7-16e+18e e-4z 4 24k-I k-.1. 2z-16e 3z+7 = Z ( ) 36 CZz 1-17. 22k-4+ J z . 1-e 9 k=3 9

65 For a.= 32 ' b= 16 ,

1 31-64e-z+66e 2z-64e-3z +31e4z 32 3 C z+ ce 180 1-e-4z - 9

co (30) 16 - [l_SZzk_6+Z4k6I z2k-1 45 >:c k=4

It is sufficient to consider even order identities after this point of the analysis, so the next identity with undetermined multi- pliers goes as

-z -2z2z+2e 3z+2ez+2e 5z+ez 2z+2ez+e-bz 1+2e +2e -a 1+2e -6z -6z 1-e 1-e (31) -3z)2 -6z 00 +e 1 +e +b (1 -c 2k-1 z2k-1. -6z -6z L1-22k-1 2k-1 1-e 1-e k=o

The open end requirement is 1-a+b-c-0

= 1 - 22k-la + 32k-lb 62k-lc 0

2k+1 2k+1 1 -22k4-1 a +3 b - 6 c =0,k =1,2,..,

It is not too practicable to solve for arbitrary values of k. For

25 15 1 k = 1 the solution is a = and Eq. (31) takes 11 ' b 11 ' c 11 ' the form 10

-z -2z 5z 3 11e 2Z+26e 3z-14e-4z+lle ( 33) 10 1-e-6z GO 1 33 25 2k-1 15 2k-1 1 2k-1, 2k-1 = 1- .+ - z . z + 10 11 -. 11 11 J k=3

From the closed end formulas the requirement is that

2k -la 2k -lb 62k -1 1 -2 a + b _ 6 c =0

(34) 22k +1a +1b 62k +1c + 32k c = 0

- 22k +3a +3b 62k +3c 1 + 32k = 0, k = 1, 2, 3, .. . . The solution is at once

(35) 7 ? 1 a = b = , k = 1, 2, 22k+1 ' 32k+1 ' c 62k+1 ...

7 For a = Eq. (31) takes the form b = Z7 , c = 216 ,

2 3Z+27e 1 41+216e-z+27e +272e 4z+216e 5z+41e-6z 140 1 - e-6z

(36) GO 1 54 2k-4 2k-4 2k-4 2k-1 C2k[1- 7 . 2 +7.3 -6 ]z + k=4

7 7 1 For a = , b 32 - 243 ' c 7776 '

2z+8000e 1 3149+7776e-z+075e 3z+6075e4z+7776e 5z+3149e-6z 7000 e-6z 1 e

(37) oo k+"( 32k-6 62k--6), Z k.1 7.22 . z+ 25 C2z+ 875 C 2kt1 k=5 J 11

7 7 1 For a = 128 ' b - 2187 ' c = 279936

3z 1 132761+ 279936eß+264627e-2z +280832e3z +264627e -4z 272580 -6z 1 -e

(38) -5z -6z +279936e 5z+132761e-6z - 6z 1 -e

0o 1 823 216 3 23328 2k-8 2k-8 2k-8 2k-1 +22715 2 +7 3 -6 ]z z+3245 C2z+649 C C2k[1=7 k=6

The formula for differentiation would result if it be required that

b 1 -2 + - 6 = 0

1 22k +1a +1b 62k +1c (39) - + 32k = 0

2k+3 2k+3 2k+3 1-2 a+3 b- 6 c=0, k=1,2,3,......

41 23 7 For k = 1 the solution is a= 16' b = 27' c= 432' andEg. (31) takes the form

5z+157e-6z 1 157-432e z+675e 2z-800e 3z+675e-4z-432e 720 -6z

(40) 00 27 22k = CZz [1-41. +23.32k-4 7 6 =4., 2k-1. 4 5 Z. z k =o To obtain the Gould- Squire or second Euler -MacLaurin formula begin with the identity 12

3z -z -z -2-2z 2z 1+2e2 +2e +2e +e (1+e-z)2 1+e a +b

2z - 1-e 1-e-2z 1-e

(41) 00 z 2k-1 2 1-22k-la+24k-2 - 1i ](2) k =o

For no over - lapping the open end condition is required; also, the

middle term on the left must be zero so the requirement on the coefficients is that

1 -a +b =0 (42) 2 -2a =0

The solution is at once a = 1, b = 0, which leads to

z 3z oo 2 (43) e +e 2 2k-1 (z)2k-1 2 -1] (2) 2z z C2k{ 1 -e k=1

The negative sign before the series is quite significant. It is well- known that the Bernoulli polynomials are defined by

zt k z e (44) _ (t) . ! ez - 1 k= o and

1 (45) B (1) _ -(1 - )B k= 1,2,3 . . . 2k 2 22k-1 2k'

Thus Eq. (43) may be written in the form '13

z 3z B ( 46) e 2+e 1+ 2k( 2) 2k-1 2= z -2z z + (2k)! 1-e k=1 A second form of this type results if it be required that

1 - a +b = 0 (47) 2 -2a = -4.

Solution is at once a = 3, b = 2, and Eq. (41) reduces to

z 3z - -z 2 e -2e +e 2 2 4k-1 2k-1 z 2k-1 ( 48) 3 = C2z + C2k[2 -3 2 +1] ( 2) . -2z 3 1-e k=2 14

EULER -MACLAURIN FORMULAS

It is quite simple to pass from the identities of the previous section to the various forms of the Euler- MacLaurin formulas. The fundamental idea is that if F(x) and f(s) are suitably restricted so that they may be regarded as a Laplace transform pair then

0, -ooh .

An identity based on that of Eq. (2) is at once

- sh co sh)f(s)_l+e-sh(1-e sh)f(s) 2k-1(1-e-sh)f( (50) (l+e e C2k( s s) , 2 1-e s k=o and inversion yields

co 2 ('x 2k-1 2k-1) ( 2k-1) (51) F(x)+F(x-h)=h F(y)dy+2 C2kh Ix (x)-F (x-h)] . xh k=1

It is convenient to set

(52) x= a +(n+ 1)h and sum over n from 0 to N -1, N = 1, 2, ... , obtaining

N a+Nh F(a h) +F(a) F(a +nh) = y F(y) dy+ F(a nLL=o n=o

co C2kh2k-1rF( 2k-1)(a+Nh) F( 2k 1)(a)] ( 53) + k= 1 15

This is the classical form of the Euler -MacLaurin sum formula and was discussed at length by Gould and Squire. The second form depends on the identity of Eq. (43); begin with

sh 3sh sh 3sh 2 2 2+e- e 2sh)f(e) e +e 2) f( s)= ( _2sh ( 1-e 1-e (54)

1 -. = C 2 k(1 2k-1)(sh)(i-e)f(s) k=1 2 and inversion leads to

x co 1 1 2k-1( 2k-1) 4-12k--R- ( 55) 2h)]. F(x-Z)+F(.x--32)= S SF(y) dy-C2k(1- L h .. 2k-1 x-2h k=1 2 It is convenient to set (56) x=a+2(n+l)h. then sum over n from 0 to N -1, N = 1, 2, ... A slight change of notation on the left side of the resulting identity leads to

2N -1 a +2Nh °O 1 F( 2k. pa (57) F[ a+(n +Z)h] _ +2Nh) F2k(á3]. Ç F(y)dy- C2k 21c 1)h k n=o a k =1 2

This is the form which appears in Hildebrand. If a is replaced by - Eq. (57) a 2 may be written as 16

2N-1 a+( 2N- 2)h 1F( a+nh) = F( y) dy h n=o a- h

00 1 2k-1 ¡ 2k-1) 1 ( 2k-1) h ) [ )h [a+(2N 2)h] (a- )> > ( 58) - C 2k(1 F k=1 2 which is equivalent to Eq. (1.8) of Gould and Squire. Both these formulas imply an even number of terms in the series on the left. The summation of terms with alternating signs depends on the implementation of the identities of Eqs. (9) and (48). Begin with Eq. (9) in the form 2sh -sh -2sh 1 - -2sh (1 - _ (1-e )f( s) 1- e -2sh

(59) 00 22k-1)( sh)2k 1( 2 e Zsh)f( s) ; k=1 inversion yields m 2k 2k-1 ( 2k-1) ( 2k-1) (60) F(x)-2F(xh)+F(x-2h) = 2 ) C 2 -1)h [F (x)-F (x-2h)]. 2k( k==1 Let

(61) x = a + 2(n+l)h then summation with respect to n from 0 to N -1, N =1, 2, ... , leads to 2N n (-1) F(a+nh)= F(a+2Nh) + F(a) n=o 2 co (62) 22k 1)h2k-1[F( 2k-1)(a+2Nh) 2k-1)(a)]. C 2k F( k=1 17

This result is proposed as an exercise by Knopp [9, p. 553] and is written out by Hildebrand [ 8, p. 157] for N- ao . It should be checked out for the simple polynomial case F(x) = xp, p =0, 1, 2, ..

For p = 0 Eq. (62) is obviously true, and the case p = 1 yields at once 2N (63) (-1)n(a+nh) = a+Nh ; n=o the case p = 2 yields 2N (a 2 Nh)2+a2 h(a Nh-a) (64) (-1)n(a+nh)2 Z á +Nh(2a+h)+2N2h2. n=o

The special situation a = h = 1 is of interest; final forms are

)n( 1 +n) 2 = (1 +N)( 1+2 N) , n=o

2N (65) (-1)n(l+n)3 = (1+N)i1+4N) , n=o

2N n (-1) ( l+n)4 = ( 1+N)(1+2N)(1+6N+4N2) . n=o

These results may be checked out by induction or by any of several other methods. In his most recently published work Polya [ 11, p.

60 -98] has discussed N (1+n)p, n=o 18 p a positive integer, at considerable length but does not include the corresponding formulas for alternating sign. For p taking negative integer values the series representation is no longer finite. That is, for p= -q, q= 1,2, ... ,

( q) 2k-1 (66) F( 2k-1)(x) = q+2k-1 x where

(67) (q)0 = 1, (q)n = q(q+1)(q +Z) (q +n -1).

Thus, for h = 1 and a understood to be a positive integer, Eq. (62) yields

2N 2N+a n+a \ (-1)n = 1 1 1 /f n ó(a+n)q n=a ng 2 (a+2N)q aq (68) CO 2k 1 1 C 2 -1)(q) [ 2k-1 q+2k-1 1 ' k=1 (a+2N)g+2k-1 a A short table may be quickly constructed to show the nature of this approximation. Let

oo 1)n+a 1 1 (69) Z _îl n + n=a = 2a 4a2 and 00 -1)n+a 1 1 1 2 A ' L n 2 n=a 2a 4a2 8a4 where Al includes only the first term of the series of Eq. (68) while A2 includes the first two terms. Since In 2 = O. 6931471 19 five place accuracy is maintained.

a exact Al A2

1 .69315 . 75000 .62500

2 . 30685 . 31250 .30469

3 . 19315 . 19444 . 19290

4 . 14019 . 14062 . 14013

5 . 10981 . 11000 . 10980

10 . 052488 . 052500 . 052487

Following the lead of Gould and Squire it seems that another formula might be available for this type of asymptotic expansion.

Begin with the identity which depends on Eq. (48),

sh 3sh -sh -3sh 2 -sh 2 e 2- 2e -sh +e 2 -2sh (e -2e +e )f(s) _ -2sh (1 -e )f(s) 1-e (71) co 2k-1 4k-1 2k-1 sh 2sh_ = C2k[2 -3.2 +11(-2) (1-e f(s), k=1 inversion yields

F(x- Z) - 2F(x-h)+F(x- Zh)

(72) co 2k-1 _12k-1)(x) 1_12k-1) C2k[24k1_3 22k-1+1](Z) [ k =1 and it is convenient to set

(73) x = a +(n +2)h. 20

Summation with respect to n from 0 to N -1 yields 2N F(a+Nh) + F(a) (-1) nF(a+ 2 ) _ n=o

(7 4) 1 24k-1-3, 22k-1+1](-)h2k-1 E j 2k-1) [ a +2 2 }.( N+ 21)h] 1

2k-1) I 2k-1) h (2k-1) + [ aN- 2)h]-F(" (a+2)-F (a- 2)

Note that for the trivial case N = 1 the two middle terms in the bracket on the right to h add up zero. If 2 is replaced by there results 2N F( a+21Vh) + F(a) ) ( -1) nF( a+nh) = n=o 2 2N (75) 22k-1+ h2 k-1 2k-1)la+(2N+1)11] 2 l C[24k-1-3 IF =1

+F +(2N-1)h] F(2k-1)(a+h)-F(2k-1)(ahj

This should be checked out for, say, F(x) = x4, a = h = 1. That is 2N 4 (1 +2N)4 +1 +(2N +2N)3 +(2N)3 -23 (2 +2N) +2N -2 = -7 2 2 4 n= o (76) = ( 1 +N)( 1 +2N)( 1 +6N+4N2) , which agrees with Eq. (65). For F(x) = x -q, h = 1, a an integer

> 0, the formula of Eq. (75) yields 21

2N n 2N+a n+a / (-1) (-1) 1 1 1 (77) + n9 I\ 2 no( a+n)q n a (a+2N)g aq

1 4k-1 2k--1 [ 1 + 1 1 1 2112 2 ,ir_1 k-(a+1+ZN)q+2k-1(a. l! -1 -1 -1 k=1 1+2N) k (a+l) (a-i)k

This blows up for a = 1, which discourages further study along this line although it is not too bad an approximation for q = 1, a > 5. As example of a Frank -line inversion of a Laplace transform consider the pair

sinh\rs -(n+ z)2Tr2t (78) f(s)= 3/2 ' F(t)-_ 1 - 2 2 s coshs r ( 2n+1) where F(t) represents the variables separable form of the solution of a problem proposed by Churchill [ 3, p. 217] . Clearly F(t) be- /7 haves like teas t -0+ and is asymptotic to unity from below Tr as t co. The geometric series expansion takes the form

1_ e-2s 1 1-2(e-2s-e-Ss+e-6s (79) f(s)= +e -e 8 =' s3/2 s3/2(l+e-2s ) for a = h = 2Nrs and F(x) = e -x, Eq. (62) yields

00 n -2(n+1)s= 1 -2s (80) (-1) e 2e + 1)(ZNrs)2k-le-ZNrs kLLL=1 C2k(Z2k n=o k=1

00 so 1 1-e-24-s-2e- 2Nrs 2k 2k -1 f(s) 3/2 11-e -Ze C2k(2 =1 }( 2s) s (81) - k=1 -2Nrs -2Nrs -ZNrs 2 3 1-e e 1 2s 17s +e 682s 3/2 - s 3 15 + 315 31185 s 22

The transform pairs here are not too well tabulated. The iterated error function may be defined by

2 0 ('00 -y ï erfc x = erfc x = \ e dy , JX (82)

oo inerfc x= in- lerfc y dy, n> 1 , X and by inductive methods it is easy to establish that

n -xt\rs n x e (4t)2i.erfc , > (83) 271 n 0, -E+1 s is a transform pair. Thus a second form of the F(t) under discussion, suitable for small values of t , is

1)n (84) F(t) = 2 _.l + ierfc nLLL=1 +Tt n=1

To multiply a suitably restricted transform by s is equivalent to taking the negative derivative of the object function with respect to t, so the continuation of the table of transforms goes as 23

x 2 x 4t -x./ s e e , 2\/Trt3

2 x. 2 ---x x 4t ,se -xN s 4iT rt- ( 2t 5) (85) x_2 x x4 10x2 4t s2e-xs + 30)e t 16 Trt7 ( 2t 2

x 2 x x6 21x4 210x2 4t - 420)e. 3-x\rs t ,s e 6 Trt9 2t3 t2

That is, the F(t) defined by Eq. (78) and by Eq. (84) is also represented asymptotically by

1 t t F(t) = 2 - 2t ierfc 1- erfc r Nit 3 15t t - 3) niTrt

17 8 40 341 32 336 840 (86) +. 2 ( - + 30) - ( - + - 420)+.. 2 t 2 t 2520t t 498960t t t

1 -1 It = 2 t 1 2 e [ 1 1 29 13 /--(1 -e )+erfc + / +- + + +, ^r 3 6 lOt 840"t2 1512tá 11 a form well adapted to computation by means of the table in Carslaw and Jaeger [ 2, p. 373] . It is at once clear that for small positive t values of t F(t) behaves like the parabola 2 , and for large values of t F(t) is asymptotic to unity from below. 24

INTEGRATION PROCEDURES

Such familiar integration formulas as Simpson's rule, the three - eighths rule, the Milne predictor- corrector formulas, are all readily obtained from the various Bernoulli identities of the first section. For example, Eq. (6) yields 1-e-2sh)f( -sh+e-2sh) 1 1+4e -sh-2sh s) 3(1+4e f( s)= 3 -2sh

(87) C2k(22k-2-1)(sh)2k1(1-e = 2sh)f(s) 2sh)f(s); sh(1-e 3 k=2 and inversion yields the Simpson rule in the form

3 [F(x)+4F(x-h)+F(x-2h)] = h J F(y) dy x-2h (88)

2k-2 2k-1 (2k 1)(x) (2k--1)1)(x- Cr(2 -1)h [F (x)-F 2h)] k= 2

The infinite series represents the successive correction at the two endpoints. A similar development of Eq. (7) leads to

[7F(x)+16F(x-h)+7F(x-2h)]= F(y) (x-2h)] 15 dy+15[F h

(89) co 16 2k-4 2k-1 (2k-1) (2k-1) [F (x)-F (x-2h)] , - 15 C2k(2 -1)h

kkk =3 25 a form which is mentioned by Milne [ 10,p. 78]. As another example note that Eq. (24) yields

Ç7F(x)+3ZF(x-h)+lZF(x-2h)+3ZF(x-3h)+7F(x-4h)]= \ F(y) dy x-4h (90) 00 64 4+ 24k- 6]hZk- 1[F,(2k- 1)(x)- 2k-1)(x C 1 -5 22k- 2h)]. + 45 k=3 which is also mentioned by Milne [ 10, p. 48].

A different approach to the various integration formulas and their respective error terms is to begin with a set of even and odd periodic polynomials, expressed in terms of their Fourier series,

x r- cós 2nTr h P2r(x, h) = 2( - r > 1, Zr n= (2nTr) (91) r- sin 2nTr h h)=2( - r > 0. P2r+1(x, +1 n= (2

The period is (0, h); Knopp [ 9, p. 522 -523]_ lists the polynomials defined over the fundamental period as 26

x 1 x P1(x,h) = - Z = +B1 h

2 B x2 x 1 P x, h) = _ + Bl. h + IT + 12 h2 2 2h 2h (92) 3 2 3 x 2 Btx = + B P3(x'h) x3 12 1 6h 4h 6h3 2h2+ 2h

B2x2 4 3 2 4 B x x x x 2 B4 P4(x, h) = 4- + X 2- 1 - 4+B h3 + 4h2 + 24 24h 12h 24h 720 24h

r -kBk Pr(x, h) r' k/ k= o

A symbolic notation is common,

(93) Pr(x,h) = 1 (h+ + B , r r. h where, of course, Bk = Bk. Note that B1 = - 2 and Bk = 0 for all other odd values of k. Also, for all values of r,

(94) hPr(x' h) = Pr-1(x, h) The usual form of the Euler -MacLaurin formula may be obtained by ordinary integration by parts. For all values of x,P1(x,h) = h - [h] - 2 , where [x] = n, n < x< n + 1, 27 so n+l)h n+l)h P1(x, h) F(x) dx = J (h - n - Z) F(x) dx J 1 nh nh (95) (n+l)h (n+l)h - - F(x)dxl)h =(h -n Z)F(x) h nh nh That is,

('(n+l)h ('(n+l)h (96) 2[F[(n+l)h] +F(nh)] = \ F(x) dx + \ P1(x, h) F (x) dx. h nh nh

Summation with respect to n from 0 to N-1 leads to

N Nh Nh F(Nh2 + F(0) (' (97) F(nh) = F(x) dx + +\ P1(x, h) F(x) dx. h 0+ n=o This may be compared with Eq. (53), except that the a of that formula is taken to be zero. This amounts to a choice of the origin of the coordinate system. Further integration by parts yields

Nh Nh Nh \ P1(x, h) F (x) dx = hP2(x, h)Fi(x) - h C P2(x, h)F (x)dx p 0 0 (98) Nh (' Nh = [ hP2(x, h)F (x)-h2P(x, h)F(x)] + h2 \ P3(x, h) Fa(( x)dx. O 0

Since the odd degree periodic functions are assigned the value zero at the cross -over points and the even order functions take the value

Bk /k! , the extension of Eq. (97) yields 28

N B2 F(Nh) +F(0) ) F(nh) = J F(x) dx + + [F(Nh)-FF(0)] h 2 n=o -0

(99) Nh

+ h2 P3(x, h) F(x) dx . 0

Such integration by parts may be extended as far as desired, so the final form of the Euler -MacLaurin formula may be written as

N Nh K 2k-1)(Nh) 2k-1O0)] /F(nh)= F(x)dx +F(Nh)+F(0) + C h2k-1[F( h á 2 2k n=o k=1 (100) f' Nh + h2K { P2K+1(x' h)F(2K+1)(x) dx. o

To arrive at the Simpson integration formula one need only write (for N = 2, K = 1)

2h F(2h) + 2F( h) + F( 0) = F(x) dx + 2C 2h[ F( 2h) -FY( 0)] h

(101) ( 2h + 2h2 j P3(x, h) F(x) dx , o and (for N = 1, K = 1, h taken to be twice as large)

(' 2h 2h (102) F(2h)+F(0)= h J F(x)dx+4C2h[Fl(2h)-F(0)] +8h P3(x, 2h)F(x) dx; 3 0 0 multiplication of the second form by 1/2 and subtraction from the 29 first yields

2h 3 2h (' tit (103) = F(x) dx + 43 J[ P3(x, h)-2P3(x, 3.[F(2h)+4F(h)+F(0)] S :2h)]F (x)dx. 0 0 It is sometimes better to reduce each term of the error integral to an integration over the fundamental period. That is,

2h (' 2h 2h \ P3(x, h)F(x) dx = ,\ P3(x, h)F(x) dx + \ P3(x, h) F(x+h) dx J J -0 OJ 0 (104) 2h = ,11PP3(x, h)[ Fm(x) + F(x+h)] dx , 3 0 and the Simpson rule may be written as

2h ( 3 [F(2h) + 4F(h) +F(0)] = F(x) dx 0

3 h 2h +43 [SI P3(x, h)[F(x)+F(x+h)]dx-2 \ P3(x, ,2h)F(x)dx. 0 0 4 It is easy to illustrate with, say, F(x) = x . 5 h 2h ('2h 3 2 = J x4dx +32h S P3(x, h)( - 2 \ P3(x, 2h)x dx 3 2x+h)dx 0 0 0 2h h 2 -0\ x4dx +32h3 [ h(2x+h)P4(x, h)-h P h}] 5(x' 0 1 1 [- -2[2hxP4(x, 2h)-4h2P5(x, 2h)] 2h 0 2 2 (' h = x4dx +32h3 0 _ 360 +9 I 30

That is, 2h 5 (' 20h [ x4dx 1-25 ] 0 and the result of applying the ordinary Simpson rule for this special case is four percent too high.

It is clear that an extended form of Simpson's rule is available which would be comparable to Eq. (298) of Knopp [ 9, p. 524] . Re- write Eq. (100) with N replaced by 2 , h replaced by 2h, multi-

1 ply by 2 , subtract from the original form, arriving finally at

F(0)+4F(h)+2F(2h)+ 4F(3h)+. . . .. +4F(N-1)h]+F(Nh)]= F(x)dx 0'1

K (106) C - 2k-2 2k (2k-1) (2k-1) I 2k k=2

Nh 4h2k+1 2K-1 ( 2K+1) h) 2 2h)] F (x) + 3 [P2K+1(x, dx. 0' P2K+1(x'

Knopp [ 9, p. 531 -534] devotes several pages to the discussion of the behavior of the P (x, h), with illustrations of how best to 2K +1 evaluate the remainder term. If F(x) is defined for all x > 0 and,

together with all its derivatives, tends - monotonely to zero as x increases, the error term may be quite easily estimated since the alternating sign associated with the C2k would imply an alternating series. 31

BIBLIOGRAPHY

1. Adams, J. C. Table of the first sixty -two numbers of Bernoulli. Journal fur die Reine und Angewandte Mathematik 85:269 -272. 18 78 .

2. Car slaw, H. S. and J. C. Jaeger. Conduction of heat in solids. Oxford, England, Oxford University Press, 1947. 386 p.

3. Churchill, Ruel V. Operational Mathematics. 2d ed. New York, N. Y. , McGraw -Hill, 1958. 337 p.

4. Erdelyi, Arthur, et al. Higher transcendental functions, vol. I. New York, N. Y. , McGraw -Hill, 1953. 302 p.

5. Fort, Tomlinson, Finite differences and difference equations in the real domain. Oxford, England, Oxford University Press, 1948. 251 p.

6. Frank, Irving. An application of the Euler -MacLaurin sum formula to operational mathematics. Quarterly of Ap- plied Mathematics 20:89 -91. 1962.

7. Gould, Henry W. and William Squire. MacLaurin's second formula and its generalizations. American Mathemati- cal Monthly 70:44-52. 1963.

8. Hildebrand, F. B. Introduction to numerical analysis. New York, N. Y. , McGraw -Hill, 1956. 511 p.

9. Knopp, Konrad. Theory and application of infinite series. 2d English ed. New York, N. Y. , Hafner, 1947. 563 p.

10. Milne, William E. Numerical solution of differential equations. New York, N. Y. , John Wiley, 1953. 275 p.

11. Polya, George. Mathematical discovery, vol. I. New York, N. Y. , John Wiley, 1962. 216 p.