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, differentia- tion, 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 ex- tended 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 by 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 inver- sion 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 for- mulas 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 unde- termined parameter, and of others introduced below, the develop- ment of (1) an open end integration formula, (2) a closed end inte- gration 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 formu- la but it does lead at once to the well -known Simpson rule and vari- ations 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 oo 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 - .