20.5 Numerical integration 577
wheref
0′,f
n′,f
0′′′, f
n′′′, =are the odd-order derivatives off(x)evaluated at the
endpoints, and the numbers =are called Bernoulli
numbers. Whenf(x)is a polynomial of degree nthe expansion inh
2terminates after
a finite number of terms because all derivatives after the nth are zero. The formula
(20.36) is then exact. In other cases, the expansion is not usually a convergent series
for any value of hbecause the Bernoulli numbers B
n, after the first few, increase
rapidly in magnitude as nincreases. It is instead a special kind of expansion called an
asymptotic expansion, and it has the property that when truncated at any point the
error is less than the magnitude of the last included term.
Apart from its use for the analysis of error in numerical integration, the Euler–
MacLaurin formula has some more direct applications in the physical sciences. The
formula can be used for evaluating sums of the type over a set of integral
values of m. Thus, puttingh 1 = 11 and inserting the values of the Bernoulli numbers,
(20.36) can be rearranged as
(20.37)
EXAMPLE 20.12Evaluate
Forf(x) 1 = 112 (a 1 + 1 x)
2,
and the function and all its derivaties tend to 0 asx 1 → 1 ∞. Then
to 6 significant figure. We note that a very large number of terms of the sum over m
would be required to obtain just the first term of the expansion.
0 Exercises 24, 25
=0 105166.
1
10
0 1 0 005 0 00016666 0 000000
20()
....
=+ + −
=∑
m
m∞3 33 0 000000002+−.
=+ + − + −
11
2
1
6
1
30
1
42
23 5 7a
aa a a11
2
00
1
720
20()
() () ()
am
fxdx f f
mab=+−′ +
=∑
∞Z
1
12
ff′′′′()−+()0
1
0
30240
vZ
02351 1 02 024
∞()axdxaf+=,() ,()′ =− af′′′ =− af,
v(()0 720
7=− a
110
20()+
=∑
m
m∞1
12 30240
()( ff′− ′ − f f′′′− ′′′ ) (+
nn 001
720
1
fff
nvv−−
0)
ffxdxff
mmnabn=++
=
∑001
2
Z ()
f
m∑
BB B
21641306142=, =− , = ,