Calculus: Analytic Geometry and Calculus, with Vectors

232 Integrals

over the interval - oo < x < co, except that (2)fails to hold when n is 1 or 2
and x is an integer. They all have period 1, that is, Bn(x + 1) = Bn(x) for each

1 2 3

Figure 4.395

n and x. They are all continuous except that

Bl(x), the saw-tooth function having the graph

shown in Figure 4.395, is discontinuous at the

integers. In fact, B1(x) = 0 when x is an integer

and

(4) B1(x)=x-[x]-

when x is not an integer. Show that (1) and (2) imply existence of constants

Bo, B1, B2, suc h that, when 0 < x < 1 and 0! = 1 as usual,

(5) B o(x) = 0 o

(5.1) B I(x) =Box+ B

0!1! 1!0!

(5.2) B

z

+

B1x

z(x) =

Boz

+

B2

0121 1 2!0!

(53) B 3(x) = Box- +

B1xz

+

Bzx

-}-

B3

. 013! 112! 2!1! 3!0!

(5.4) B 4(x) =

Box' + Blx3 + Bzxz + Box + B4

0!4! 1!3! 2!2! 3! 1! 4!0!

and write two more of these formulas. Because of continuity, each of these

formulas except (5.1) holds when 0 < x 5 1. The numbers Bo, B,, B2,

are the Bernoulli numbers and, when n >t 2,

(6) Bn = n!Bn(0).

Show that the above formulas can be put in the neater forms

(7) 0!Bo(x) = Bo

(7.1) 1!Bl(x) = Box + B,

(7.2) 2!Bz(x) = Boxz + 2Blx + B2

(7.3) 3!B3(x) = Box3 + 3B,xz + 3B2x + B3

(7.4) 4!B4(x) = Box4 + 4Bix3 + 6B2x2 + 4B3x + B4

involving binomial coefficients and write two more of these equations. Use (3)

to show that, when n >= 2,

Bn(1)-B.(0)= foI Bn(x) dx = 101 Bn-i(x) dx = 0

and hence

(8) Bn(1) = Bn(0)

Use (1) and (7) to show that Bo = 1. Then use (7.2) and (8) to show that

B1 = -i-. Then use (7.3) and (8) to show that B2 =. Then calculate one

or two more Bernoulli numbers. Remark: Bernoulli functions and numbers

have important applications and some people know very much about them.

It can be shown that Bo = 1, B1 = - , B2 =g, Be = 0, B4 = - 1 8, B5 = 0,