Euler-Maclaurin summation formula

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1 Physics 4 Spring 6 Euler-Maclaurin summation formula Lecture notes by M. G. Rozman Last modified: March 9, 6 Euler-Maclaurin summation formula gives an estimation of the sum N in f i) in terms of the integral N f )d and correction terms. It was discovered independently by Euler and Maclaurin and n published by Euler in 73, and by Maclaurin in 74. The presentation below follows[], [, Ch Y], [3, Ch ]. Preliminaries. Bernoulli numbers The Bernoulli numbers are rational numbers that can be defined as coefficients in the following power series epansion: e n n!. ) These numbers are important in number theory, analysis, and differential topology. Unless you are using a computer algebra system for series epansion it is not easy to find the coefficients in the right hand side of Eq. ). However, it is easy to write Taylor series for the reciprocal of the left hand side of Eq. ): e k k k! k k + )! ) k Thus, the Bernoulli numbers can be computer recurrently by equating to zero the coefficients at positive powers of in the identity e e k + )! n! k+n n,k ) B + B! + B )! +... B B +!! + B ) B +!! 3!! + B!! + B ) )!! For eample, Series[/Ep[] - ),,, 6] in Mathematica Page of

2 Physics 4 Summation of series: Euler-Maclaurin formula Spring 6 From here, B, B B, B 3 B B 6, etc. B 4) is the only non-zero Bernoulli number with an odd subscript. The first Bernoulli numbers with even subscripts are as following: B, B 6, B 4 3, B 6 4, B 8 3, B 5 66,.... 5) The first few terms in the Epansion Eq. ) are as following: e ) Preliminaries. Operators ˆD and ˆT If f ) is a good function meaning that we can apply formulas of differential calculus without reservations ), then the correspondence can be regarded as the operator of differentiation f ) f ) d f ) 7) d ˆD d d that act on the function and transforms it into derivative. Given ˆD, we can naturally define the powers of the operator of differentiation ˆD f ) ˆD ˆD f ) ) d d d f ) d d d f ) i.e. ˆD d d, 9) or in general, ˆD n dn d n. ) We can define functions of the operator of differentiation as following: if g) is a good function that can be epanded into power series, g) 8) a n n, ) then the operator function g ˆD) is defined as following. g ˆD) a n ˆD n ) Page of

3 Physics 4 Summation of series: Euler-Maclaurin formula Spring 6 Let s consider the eponential function of the differential operator: When applied to a good function f ), ˆT f ) ˆT e ˆD ˆD n ˆD n f ) ) n! The last epression in Eq. 4) is just a Taylor series for f + ). Thus, n!. 3) d n f n! d n. 4) ˆT f ) f + ). 5) and ˆT can be regarded as the shift operator. Shifting by can be considered as a composition of two shift by one operations: f + ) ˆT f + ) ˆT ˆT f ) ) ˆT f ). 6) Similarly, shifting by a positive value s can be considered as the result of operator ˆT s : f + s) ˆT s f ). 7) 3 Summation of series in terms of operator ˆD We can now formally write f + n) f ) + f + ) + f + ) +... f ) + ˆT f ) + ˆT f ) ˆT + ˆT +... ) f ) ˆT f ) f ), 8) e ˆD where the epression for the sum of geometric progression was used. Treating ˆD as an ordinary variable, and using Bernoulli numbers, we obtain the epansion: ˆD e ˆD ˆD e ˆD ˆD ˆD + n ˆD + n n! ˆD n n! ˆD n 9) The question remaining before Eq. 9) can be applied is what does ˆD mean. It is natural to assume that ˆD satisfies the relation ) ˆD ˆD f ) ˆD f ) f ). ) ˆD Page 3 of

4 Physics 4 Summation of series: Euler-Maclaurin formula Spring 6 Therefore ˆD has to be an inverse operator to differentiation, that is integration. ˆD f ) f ) d + C. ) We still need to fi an integration constant i.e. to chose the integration limits. As we see later, we obtain consistent results, if ˆD f ) f ) d, ) or Collecting the results together, ˆD f ) f ) d. 3) f + n) f ) d + f ) n n! d n f ). 4) d n 4 The Euler-Maclaurin summation formula Equation 4) is the Euler-Maclaurin summation formula. It can be rewritten for the case of a finite sum as following: N kn f k) f k) kn f k) kn kn+ kn f k + n) k n + f ) d n N n n! f k) f k) + f N) f k + N) + f N) k N d n f n) d n + f ) d + f ) d + f n) f N) + f N) n n! d n f N) d n [ f n) + f N) ] + n n! [ d n f d N n dn f ] d n n. 5) Page 4 of

5 Physics 4 Summation of series: Euler-Maclaurin formula Spring 6 5 Stirling s formula As an application of Euler-Maclaurin summation formula, let s consider the Stirling s approimation for Γn) for positive integer n. Γn + ) n!, 6) ln Γn + ) ) ln n! ln 3... n ) n ) n ln k. 7) The first two terms in Eq. 5) give us the following approimation: ln Γn + ) ) lnn!) The net correction requires more efforts. n k ln) d + ln) + lnn)) n ln) n d + lnn) n lnn) n + + lnn). 8) Let s notice first that the term with the derivatives of ln) at n in Eq. 5) are proportional to negative powers of n and thus as n. On the other hand, the sum of the term with the derivatives of ln) at is a constant independent of n. Thus, n ) n [ ln Γn + ) ln n! ln + ln n + lnc) ln C n ) n ] n, 9) e e or Γn + ) n! C n ) n n Cn n+ e n. 3) e In order to find the constant in Eq. 3), we are going to use the duplication formula for Gamma function, Eq. 47). We first rewrite eq. 47) as following: Γn + ) n Γn + ) Γ n + ), 3) π or n)! n n! Γ n + ), 3) π Let s accept without proof that Eq. 3) that we derived for integer n works also for any large positive argument. Indeed, if Eq. 3) is a good approimation for Γn) and for Γn + ) it is reasonable to assume that it works in between n and n +. Therefore, Γ n + ) Γ n ) + C n ) n e n+. 33) Page 5 of

6 Physics 4 Summation of series: Euler-Maclaurin formula Spring 6 For n, Thus, n ) n n n ) n n Γ n n e. 34) n + ) Cn n e n. 35) Substituting Eq. 3), 35) into Eq. 3), we obtain: C n) n+ e n n π Cn n+ e n Cn n e n. 36) After simplification, Finally, C π. 37) Γn + ) n! n ) n πn πn n+ e n. e 38) 7 6 Figure : Stirling s approimation Eq. 38), solid line, compared with Gamma function, dashed line, and factorial, circular markers Eamples Eample. Lets consider the following sum: Sα, k) e αn ) k, α >, k >. 39) n Page 6 of

7 Physics 4 Summation of series: Euler-Maclaurin formula Spring 6 The case of small α, α, is most difficult for a numerical summation, since many terms need to be added in the sum Eq. 39). Small α is where the Euler-Maclaurin approimation works the best. Sα, k) Introduction of a new integration variable, u α k transforms the integral as following: so that k α k e α ) k d e αk d. 4) u α ) k d k α k u k du, 4) e u u k u k du α k ) Γ, 4) k k Sα, k) α k ) Γ. 43) k k The approimation Eq. 43) is compared with the results of numerical calculations in Fig. and Fig Figure : Euler-Maclaurin approimation Eq. 43), dashed line, compared with numerical value of the sum Sα,), solid line. Sa) Sa) n e an numerics Euler-Maclaurin approimation a Eample. Euler-Maclaurin summation formula can produce eact epression for the sum if f ) is a polynomial. Indeed, in this case only finite number of derivatives of f ) is non zero. Thus there is only a finite number of correction terms in Eq. 5). Page 7 of

8 Physics 4 Summation of series: Euler-Maclaurin formula Spring 6 9 Figure 3: Euler-Maclaurin approimation Eq. 43), dashed line, compared with numerical value of the sum Sα,), solid line. Sa) Sa) n e an4 numerics Euler-Maclaurin approimation a Let s consider the following sum: S 3 Euler-Maclaurin epression for the sum is eactly as following, S 3 n 4 3 d + n 4 ) + where we used B 6. After some algebra, n k 3. 44) k n 3 + ) + B n 3 + ) + 4 3n 3 ) + B 4 4! 6 6) n ), 45) which is indeed the correct result. S 3 4 n n + ), 46) Appendi A. Duplication formula for Gamma function The duplication formula for the Gamma function is Γz) z Γz) Γ z + ) π 47) It is also called the Legendre duplication formula. Page 8 of

9 Physics 4 Summation of series: Euler-Maclaurin formula Spring 6 We start from the definition of Beta function, Bz, z). Bz, z) z ) z d. 48) Let s change the integration variable to t, +t, so that t and d dt. This transforms Eq. 48) into Bz, z) z t) z + t) z dt z t ) z dt. 49) Changing the integration variable in the last integral to u t, so that u and dt u, we transform the integral to i.e. In terms of Gamma function, so that Bz, z) z u u) z du z B Bz, z) z B B Bz, z) ), z Γz)Γz) Γz) ), z, 5) ), z. 5) Γz) Γz), 5) Γz) Γ ) Γz) Γ ), 53) z + Γ ) Γz) z Γ ). 54) z + Rearranging, and using the value of Γ/) π, we see that Γz) z Γz)Γ z + ), 55) π which is the duplication formula. References [] Sergey Sadov. Euler s derivation of the euler-maclaurin formula, 3. URL math.mun.ca/ sergey/winter3/m33/asst8/eumac.pdf. [Online; accessed 6-3-4]. Page 9 of

10 Physics 4 Summation of series: Euler-Maclaurin formula Spring 6 [] Sanjoy Mahajan. Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving. MIT Press,. [3] Hung Cheng. Advanced Analytic Methods in Applied Mathematics, Science, and Engineering. LuBan Press, 7. Page of