CiteSeerX — A note on the Euler-Maclaurin Sum formula

TR/01/93 March 1993

A note on the Euler-Maclaurin Sum formula

A. D. Rawlins

w9258496

A note on the Euler-Maclaurin Sum formula

by A. D. Rawlins

Department of Mathematics & Statistics Brunel University

Uxbridge Middlesex UB8 3PH

Abstract

In this note we give a real variable approach for calculating the constant term that arises in the application of the Euler-Maclaurin expansion for a special class of series of the form

∑

= n

1 r

f(r) asn→ ∞.

In particular the method is used to derive the approximate summation of the expression where ℓ is a non negative integer.

∑

= n

1 r

r l^l nr,

-2- Introduction

A problem of frequent occurrence in analysis and applied mathematics is to find an approximate expression for sums of the form

∑

=

∞

→

= ⁿ

1 r

n f(r), as n S

This is particularly important when f(x) is a slowly varying function of x, when the above expression for Sn would be useless for calculating . An effective mathematical method for dealing with this type of problem is the Euler-Maclaurin summation formula. One form of this summation formula gives an estimate for the sum S

n lim n_→∞S

n, by the integral:

dr,

nf(r)

∫

1

with correction terms, involving a constant, and the values of f(r) and its odd derivatives at t = n. A very good and comprehensive treatment of the Euler-Maclaurin summation formula is given in the book by Olver[l], chapter 8. The evaluation of the constant term requires some ingenuity, especially when f is real with only a finite number of continuous derivatives, (that is, f(x) cannot be analytically continued off the the real x-axis).

We shall describe a method for the evaluation of the constant term which seems more direct than that usually used in text books. The method works when high enough derivatives of f can be expressed in inverse powers of x² or x³. The method uses the periodicity of the Bernoulli polynomials B2s(x-[x]) and the fact that

n) . (x 1) 1

dx nΓΓ(

d

0

n 2

2

2

∑

^∞

= +

= l +

where Г(x) is the Gamma function, O1ver[l]. As an application we consider the sum ℓ a non- negative integer.

, nr

n r

1

∑

r

= ^ll

Let f(x) have 2m continuous derivatives f^(2m) (x) for x≥1, and let f^(2m-1)(x) 0, f>< ^(2m)(x) <> 0, f^(2m-1)(x)→0 as x→∞, then the Euler-Maclaurin sum formula gives

∫ ∑

∑

=

−

−

=

+ + +

+

= ^m

1 s

1 2m 2m

1) 2s (2s n

1 n

1 r

(n)), 0(f

C (n) (2s)!f

f(n) B 2 f(x)dx 1

f(r) (1)

m = 1,2,..., see Olver [1].

In the above expression on the right hand side of the equality sign, all the terms before the constant C2m increase with n, C2m = 0(1), and the order term is o(l) as n→∞. The constant term C2m is given by the expression

∑

⁻

∫

^∞

=

− −

−

= 1

2m (2m) 1)

m (2s 1 s

2s

2m f (x)dx

(2m)!

[x]) ((x (1) B

(2s)!f f(1) B

2

C 1 (2)

-3-

∫ ( ) ∑

∑

^∞

=

−

=

+ +

−

−

= ¹

0 r 0

) m 2 m (

) 2 1 s 2 m (

1 s

s

2 f (x r 1)dx,

)!

m 2 (

x ) B

1 ( )!f s 2 ( ) B 1 ( 2f

1 (3)

provided the infinite series in (3) converges uniformly.

Evaluation of the constant term C2m.

For the applications we have in mind we will need to consider two situations.

(i) f^(2m) (x) = a(2m)x^-2 , a(2m) independent of x.

Then we can write (3) in the form

∑ ∫ ∑

=

∞

=

−

+

− +

−

= ^m

1 s

1

0 2m r 0 2

1) 2s (2s

2m dx,

1) r (x (x) 1 (2m)! B

(1) a(m) (2s)!f

f(1) B 2 C 1

dx 1) (x dx n

(x) d (2m)! B

(1) a(m) (2s)!f

f(1) B 2

1 ^m

1

s 2

1 2

0 2m

1)

2s (2s − +

−

=

∑

₌ ⁻

∫

^{l Γ}

Now integrating by parts twice gives

B (x) n (1 x)dx.

2)!

(2m B a(m) (2m)!

(1) a(m) (2s)!f

f(1) B 2

C 1 ^m

1 s

1 0 2m 2 2m

1) 2s (2s

2m +

− −

−

−

=

∑

₌ ⁻

∫

^{− l Γ}

(4)

(ii) f^(2m)(x)=b(m)x^-3 , b(m) independent of x.

Then we can write (3) in the form

∑ ∫ ∑

=

∞

=

−

+

− +

−

= ^m

1 s

1

0 2m r 0 3

1) 2s (2s

2m dx,

1) r (x (x) 1 (2m)! B

(1) b(m) (2s)!f

f(1) B 2 C 1

1)dx.

(x dx n

(x) d (2m)! B

(1) b(m) (2s)!r

f(1) B 2 1

3 3 2m 1 0 1)

2s (2s

m 1 s

+ +

=

=

∑

⁻

∫

= l Γ

Now integrating by parts thrice gives

( ) ( )

2(2m 3)! ^{B (x) n (x 1) dx.}

B b(m)

! 2m 2

m (1) b

(2s)!f f(1) B

2

C 1 ^m

1 s

1 0 2m 3 2m

1) 2s (2s

2m

∑ ∫

= − − +

− −

−

−

= l Γ

(m>1) (5)

To obtain (4) and (5) we have used the results (see O1ver[l]).

n (z).

dz ) d z ( where ,...,

1 , 0 m m!

) ( ) 1 ( )

2

( ^{(m) m (m) m}_m¹₁

(m) + l Γ

− +

=

−

=

−ψ ψ

ψ

..., 2 , 1 s ), x ( sB ) x ( B (m), B ) 0 ( B

B_s = _s = _s ^'_s = _s−1 =

The integrands of the integrals appearing in the expressions (4) and (5) consist of a polynomial multiplied by ℓnГ(x+l). The smooth behaviour of these integrands over the finite range of integration (0,1) is such that they can be numerically evaluated without difficulty, and hence give a numerical value

-4-

for the constant C2m to a desired degree of accuracy. Further if integrals of the form

(6)

∫

0^{1 + =}

Γ n (x 1)dx, r 0,1,2..., x l Γ

can be evaluated in closed form, then one can obtain explicit analytic expressions for the constants C2m. The result (6) for r = 0:

∫

0^{1 + =} n(2 )⁻1, 2

dx 1 ) 1 dx (

n _l π

l Γ (7)

is well known, sometimes called Raabe's result. However, it does not seem to be known that (6) can be evaluated in closed form for non-negative integers in terms of the Riemann Zeta function ζ(z), see Olver[1]. Specifically:

∫

0^{1 + =} ₊ ⁻ ₊ 2 r

) 1 r (

1 ) 1 r ( 2

) 2 ( dx n ) 1 x ( n

x _l π

l Γ

∑

⁻

=

− + + ^{r 1}

0

k k (k 2)

) 2 (

) 2k sin(1

k)!

(r r!

4

1 ζ

π π π

∑

⁻

=

− +

− + ^{r 1}

0

k k

2 (k 2)

) 2 (

) 2k cos(1

k)!

(r r!

2 )) 2 ( n

( ζ

π π π

π γ _l

, ) 2 k ) (

2 (

) 2k cos(1

k)!

(r r!

2 1 ^{r 1}

0 k

' k 2

∑

⁼

=

− +

− ζ

π π

π (8)

r = 0,1,2….

The derivation of the result (8) is as follows:

, dx x) ( n x dx nx x dx ) 1 x ( n

x ¹

0 1 r

0 1 r

0

r

∫ ∫

∫

^{l Γ + = l + l Γ}

. (x)dx n

) x 1 r (

1 ¹

0 r 2 ⁺

∫

− +

= l Γ

We now use Kummer's Fourier Series representation for ℓnГ(x) given by

∑

^∞

= ⎭⎬⎫

⎩⎨

⎧ +

+

=

1 n

nx.

2 sin n(2π(2πn)) nπ(γ

nx 1 cos2π 2n n(2π( 1

2 ) 1 (

n _{l l} π

l Γ x

0 < x < 1.

Thus

∑ ∫

∫

^∞

=

+ + + −

= +

1 m

1 r 2 0

1 0

r x cos2 mx dx

2m 1 1)

(r 1 1) 2(r

) 1)dx n(2π

(x n

x _{l Γ} ^l π

∑

^∞_{= ⎩}^{⎨⎧ +}

∫

_⎭^⎬⎫

+

1 m

1 0

rsin2 mxdx m x

)) m 2 ( n

( π

π π γ _l

-5-

A simple application of integration by parts gives the results:

∫ ∑

⁻

= + ⎟

⎠

⎜ ⎞

⎝

⎛

= ^{r 1} −

0

k k 1

1 0

r k ,

2 .sin 1 n) (2

1 k)!

(r dx r!

nx 2 cos

x π

π π

(9)

∫

⁼⁻

∑

^r−₌¹ ₋ ^{+ ⎜}_⎝^{⎛ ⎟}_⎠^⎞

0

k k 1

1 0

r k ,

2 .cos 1 n) (2

1 k)!

(r dx 1

nx sin2

x π

π π

r=0,1…

Thus the result (8) follows with

∑ ∑

^∞

=

∞ −

=

− > = >

=

2 n 1

n

z,z 1, '(z) nn.n ,z 1. n

) z

( ζ l ^z

ζ

Thus in principle we can evaluate all terms of the expression (1) explicitly, in term of functions which are well tabulated.

Application

We shall now consider an application of the previous results to obtain an approximate expression for the sum

∑

where ℓ is an integer, and n→∞.

= n

1 r

r l^l nr

Let us consider therefore fℓ(x) = x^ℓℓn x, then

_∑ ( )

=

= −

≡ ^m

0 r

r) (m (r)

m r (m)

m m

), (x nx)D ( D nx)

(x D (x) dx f

d _{l l}

l l l

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

+ +

−

+

−

−

− − +

= −

∑

=

−

1) r m (

1)}

r 1)....(m .{m(m

r ) ( 1) m ( x nx

! ^m

1 r m r

l l

l ^l l

Γ

Γ (10)

1) . r m (

1)}

r 1)....(m {m(m

r ) x (

! (x) dx f then d 1 m

If ^m

1 r

r m

m

m

∑

=

−

+ +

−

+

−

−

− −

= +

≥l l l ^l Γ l

If we choose: m = 2m = ℓ+2, if ℓ even , m = 2m = ℓ+3, if ℓ odd , we get

even ,

)x ( a dx f(x)

d ~ ₂

2m 2m

l l ⁻

=

(11)

( )

x , odd.

b^~ l ⁻³ l

= where

2)!

(r

3)}

r 1)....(

2)(

.{(

r )

! ( ) (

a~ ²

2 r

r

−

+

− +

+

− −

=

∑

⁺

=

l l

l l

l ^l (12)

∑

⁺

= −

+

− +

+

− −

= ³

3 r

r

3)!

(r

4)}

r 2)....(

3)(

{(

r )

! ( ) (

~b l l ^l l l l (13)

-6- Hence the expressions (1) and (4) give for ℓ even

nn 2n 1 1) (

n 1 nn nr n

r ₂

1 n 1

1

r l

l l

l l ^l

l

l l +

− +

= ⁻+ ⁺

∑

=

1 2)/2 2s

(

1 s

1 2S

1 r

r

2s n

2) r 2s (

r)}

2)....(2s 1)(2s

{(2s r

) ( 2) 2s (

nn (2s)!

! B ^{− +}

+

=

−

∑ ∑

= ⎥

⎦

⎢ ⎤

⎣

⎡

+ +

−

−

−

−

− − +

+ ^l − ^l

l l

l l

Γ Γ

) 0(n

C~ ₁

2

−

+ +

+ _l (14)

Where

2 2)/2

(

1 S

1 2S

1 r

r 2S

2 ( 1)

1 2)

r 2s (

r)}

2)....(2s 1)(2s

{(2s r

) ( (2s)!

! B C~

+ +

⎥⎦

⎢ ⎤

⎣

⎡

+ +

−

−

−

−

=

∑

⁺

∑

−

=

−

+ =

l l ^l l

l Γ

2)!

(r

3)}

r 1)....(

2)(

{(

r ) ( 1) 2)(

(

B ²

2 r

r 2

−

+

−

−

−

− +

+ +

∑

⁺

=

+ l l l

l l

l l

∑

⁺

∫

=

− +

− +

− +

+ ² +

2 r

1 0 r

1)dx.

(x n (x) 2)! B

r(r

) 3)}(

r 1)....(

2)(

{(

l

l l

l l

l Γ (15)

The expression (1) and (5) give for ℓ odd

n n 2n 1 1) ( n n 1) n ( nr n

r ₂

1 n 1

1

r l

l l

l l ^l

l

l l +

− +

= +^{+ +}

∑

=

∑

⁺

∑

=

−

− −

= ⎥

⎦

⎢ ⎤

⎣

⎡

+ +

−

−

−

−

− − + + ^{( 2)/2} −

1 s

1 1 2S

2S

1 r

r

2S n

2) r 2s (

r)}

2)...(2s 1)(2s

{(2s r

) ( 2)

2s (

n n (2s)!

!^l B ^l

l l

l l

Γ Γ

) 0(n

C~ ₂

3

−

+ +

+ _l (16)

Where

∑

⁺

∑

=

−

=

+ ⎥

⎦

⎢ ⎤

⎣

⎡

+

−

−

−

− + −

= + ^{( 3)/2}

1 S

1 2S

1 r

r 2S

3 2

2) 2s (

r)}

2)....(2s 1)(2s

{(2s r

) ( ) (2s!

! B 1) (

C~ 1 ^l

l l l

l Γ

∑

⁺

= +

−

+

− +

+

−

+ + ³

3 r

r 3

3)!

(r

4)}

r 2)....(

3)(

{(

r ) ( 3)!

2(

B

! _l ^l l l l

l l

∑

^{− + +} ₋ ^{− +}

∫

⁺

+ ⁺

=

1 0 3

3 r

r

dx 1) x ( n (x) 3)! B

(r

4)}

r 2)....(

3)(

{(

r ) ( 2

1 l l l l Γ

l

l (17)

-7- As specific examples we apply (14) and (15) for (i) ℓ = 0, m = 1, giving

), 0(n C n 2 n n 1 n n n r

n ¹

n

1 r

−

=

+ + +

−

∑

^l = ^{l l}

, 1 ) 2 n(2 1 1 1)dx (x n 1

C ¹

∫

0 ^{+ = + −}

+

= l Γ l π

), 2 n(2 1l π

=

where we have used the result (8) with r = 0. This result agrees with Olver[l].

(ii) We also apply (16) and (17) for ℓ=l, m=2 giving

) 0(n C n 12 n n 1 n 2n 1 4 n n 2 n r n n

r ^{2 2 2}

n

1 r

−

=

+ + +

+

−

∑

^l = ^{l l l}

∫

^{− − +}

−

+ ¹

0 ) n (x 1)dx.

2 (x 1 4

C 1 l Γ

∫

⁺

−

− +

= ¹

0x n (x 1)dx, 2

) 1 4 n(2 1 4

1 l π l Γ

(2), 2 '

1 12

)) n(2 (γ

2 ζ π − π

= +l

where we have used the result (8) with r=0 and r=l and the fact that ζ(2)=π²⁶. This result for C agrees with Olver[l] who obtained it by a different method.

References

F.W J. Olver, Asymptotics and Special functions. Academic Press 1974.

NOT TO BE REMOVED

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