MIT OpenCourseWare http://ocw.mit.edu 8.323 Relativistic Quantum Field Theory I Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I Prof. Alan Guth March 2, 2008 LECTURE NOTES 2 NOTES ON T HE EULER-MACLAURIN SUMMATION FORMULA These notes are intended to supplement the Casimir effect problem of Problem Set 3 (2008). That calculation depended crucially on the Euler-Maclaurin summa-tion formula, which was stated without derivation. Here I will give a self-contained derivation of the Euler-Maclaurin formula. For pedagogical reasons I will first derive the formula without any reference to Bernoulli numbers, and afterward I will show that the answer can be expressed in terms of these numbers. An explicit expression will be obtained for the remainder that survives after a finite number of terms in the series are summed, and in an optional appendix I will show how to simplify this remainder to obtain the form given by Abramowitz and Stegun. The Euler-Maclaurin formula relates the sum of a function evaluated at evenly spaced points to the corresponding integral approximation, providing a systematic method of calculating corrections in terms of the derivatives of the function evalu-ated at the endpoints. Consider first a function defined on the interval −1 ≤ x ≤ 1, for which we can imagine approximating the sum of f (−1) + f(1) by the integral of the function over the interval: � 1 f(−1) + f(1) = dxf(x)+ R1 , (1) −1 where R1 represents a correction term that we want to understand. One can find an exact expression for R1 by applying an integration by parts to the integral: � 1 � 1 dxf(x)= f(−1) + f(1) − dxxf(x) , (2) −1 −1 so � 1 R1 = dxxf(x) , (3) −1 where a prime denotes a derivative with respect to x.� � � � � � � � 8.323 LECTURE NOTES 2, SPRING 2008: Euler-Maclaurin Sum Formula p. 2 1. Expansion by successive integrations by parts: We want an approximation that is useful for smooth functions f (x), and a smooth function is one for which the higher derivatives tend to be small. Therefore, if we can extract more terms in a way that leaves a remainder term that depends only on high derivatives of the function, then we have made progress. This can be accomplished by successively integrating by parts, each time differentiating f (x) and integrating the function that multiplies it. We can define a set of functions V0(x) ≡ 1 ,V1(x) ≡ x, (4) and Vn(x) ≡ dxVn−1(x) . (5) Eq. (5) is not quite well-defined, however, because each indefinite integral is defined only up to an arbitrary constant of integration. Regardless of how these constants of integration are chosen, however, one can rewrite Eq. (1) by using Eq. (3) and then successively integrating by parts: � 1 1 f(−1)+f(1) = dxf(x)+ dxV1(x) f(x) −1 −1 � 1 � 1 =dxf(x)+ V2(1)f(1) − V2(−1)f(−1) − dxV2(x) f(x) −1 −1 � 1 � � =dxf(x)+ V2(1)f(1) − V2(−1)f(−1) −1 − V3(1)f(1) − V3(−1)f(−1) + V4(1)f(1) − V4(−1)f(−1) + ... � � 1 + V2n(1)f2n−1(1) − V2n(−1)f2n−1(−1) − dxV2n(x) f2n(x) −1 � 12n � � =dxf(x)+ (−1) V(1) f−1(1) − V(−1) f−1(−1) −1 =2 � 1 − dxV2n(x) f2n(x) , −1 (6) where fn(x) denotes the nth derivative of f with respect to x.� 8.323 LECTURE NOTES 2, SPRING 2008: Euler-Maclaurin Sum Formula p. 3 2. Elimination of the od d  contributions: Eq. (6) is valid for any choice of integration constants in Eq. (5), so we can seek a choice that simplifies the result. Note that V1(x) is odd under x →−x. We can therefore choose the integration constants so that Vn(x) if n is even Vn(−x)= (7)−Vn(x)if n is odd . This even/odd requirement uniquely fixes the integration constant in Eq. (5) when n is odd, because the sum of an odd function and a constant would no longer be odd. We are still free, however, to choose the integration constants when n is even. Using the even/odd property, Eq. (6) can be simplified to � 2n1 �� � f(−1) + f(1) = dxf(x)+ (−1)V(1) f−1(1) − (−1) f−1(−1) −1 =2 (8) � 1 − dxV2n(x) f2n(x) . −1 Note that the terms in V(1) for even  involve the difference of f−1 at the two endpoints, while the terms for odd  involve the sum. Eq. (8) describes a single interval, however, and our goal is to obtain a formula valid for any number of intervals. We will do this by first generalizing Eq. (8) to apply to an arbitrary interval a ≤ x ≤ a+h, and then applying it to each interval in a succession of evenly spaced intervals. When this succession is summed, the even  terms involving the differences of the endpoints will cancel at each interior point, but the odd  terms will add. The odd  terms can therefore make a considerably more complicated contribution to the answer, but we can force them to vanish by using the remaining freedom in the choice of integration constants. When n is even in Eq. (5), we choose the integration constant so that � 1 dxVn(x) ≡ 0 . (9) −1 Eq. (9) is always true for odd functions, so it is true for all n> 0. It then follows for all n>1that � 1 Vn(1) − Vn(−1) = dxVn−1(x)=0 . (10) −1 If n is odd then Eq. (7) implies that Vn(−1) = −Vn(1), and so Vn(1) = Vn(−1) = 0 for all odd n>1 , (11)8.323 LECTURE NOTES 2, SPRING 2008: Euler-Maclaurin Sum Formula p. 4 as desired. The Vn’s are now uniquely defined. In our construction we used the antisymme-try property of Eq. (5) to fix the constant of integration for odd n, and the vanishing of the integral in Eq. (9) to fix the integration constant for even n. Eq. (9), however, holds also for odd n, and is sufficient to fix the integration constant for the odd n cases. The functions Vn(x) can therefore be defined succinctly by V0(x) ≡ 1 , (12a) � Vn(x) ≡ dxVn−1(x) , and (12b) � 1 dxVn(x) ≡ 0 (for n>0) . (12c) −1 We can use these properties to build a table for the lowest values of n: n Vn(x) Vn(1) 0 V0(x)= 1 1 1 V1(x)= x 1 2 V2(x)= x2 2 − 1 6 1 3 3 V3(x)= x3 6 − x 6 0 (13) 4 V4(x)= x4 24 − x2 12 + 7 360 − 1 45 5 V5(x)= x5 120 − x3 36 + 7x 360 0 6 V6(x)= x6 720 − x4 144 +