Calculus: Analytic Geometry and Calculus, with Vectors

12.6 Euler-Maclaurin summation formulas 641

B. are defined by the formulas

(12.63) B = n!B (0) (n > 2)

and

(12.631) Bo=1,B1= Be 0,

Bs= ,B7=0,B3=- ,B9=0,Bio= ,

Without yet knowing what is going to happen, let p and q be integers

and let f be continuous and have all of the derivatives we want to use

over the interval p< x < q. Letting k be an integer for which p 5 k

< q, we start with the simple idea that

(12.64)

fk1f(x) dx

=

fk1 f(x)Bo(x)

dx

and that we can modify the right side by integrating by parts with the

aid of (12.612). However, since B1(x) is discontinuous at k and k + 1,

we must be careful. Accordingly, we put (12.64) in the form

(12.65)

fk1 f(x)dx = lim f k+, ' f(x)Bo(x) dx.

o+

Setting

gives

u = f(x) dv = Bo(x) dx

du = f'(x) dx, v = Bj(x)

(12.651) rkk+l f(x)dx = lim jf(x)Bi(x)]k+f f

f

k+ 1-e

k+E f (x)Bi(x) dx

and hence

(12.652) f k+1f(x)dx =

f (k) + f(k + 1) -k+1 f (x)Bi(x)dx.

f

k

Adding the members of (12.652) for integer valuesof k for which p <

k < q - 1 gives the more useful identity

(12.653) f 4f(x) dx = Q f(k) -

f(p)

2

f(q)- ff'(x)B(x) dx.

p kp

Transposing gives the basic Euler-Maclaurinformula

(12.66) I f(k) = f 4f(x) dx {-f(p) 2

f(q) + Jvgf'(x)BI(x) dx

k=p p