A Proof that Euler Missed"

Evaluating the Easy Way

Tom M. Apostol

R. Ap6ry [1] was the first to prove the irrationality of

symmetry of this square about the u-axis we find

| 1 l/v~ u

~(3)= ~. 1=4s ( s

dv )du

,=1 2 - u 2 + v 2

Motivated by Ap6ry's proof, F. Beukers [2] has given

a shorter proof which uses multiple integrals to

establish the irrationality of both ~(2) and ~(3). In this

i.* [ r*-u dv

note we show that one of the double integrals con- + 4 |

sidered by Beukers, -,l/V~ 2 -

u 2 + v 2 du.

J0

J

1 fol 1

I = 1 x------y dxdy,

can be used to establish directly that ~(2) = 7r2/6. This

evaluation has been presented by the author for a

number of years in elementary calculus courses, but

does not seem to be recorded in the literature.

The relation between the foregoing integral and ~(2)

is obtained by expanding the integrand in a geometric

series and integrating term by term. Thus, we have

I 1 oo

fo fo xay

n=O

Since

X

fo dt 1 x

a 2 +t 2 - a arctan-~-

we have

u dv 1 u

2-u 2+v 2 ~ X/2-u 2

arc

tan

1 oo n oo

dy = (n +

1) 2 -~(2)"

Now we evaluate the integral another way and show

that I = ~ra/6. We simply rotate the coordinate axes

clockwise through an angle of 7r/4 radians by intro-

ducing the change of variables

U--V Uq-V

x- v~ ,y=--~--

so that 1-xy = (2-u 2 +v2)/2. The new region of

integration in the

uv-plane

is a

square with two

oppo-

site

vertices at (0,0) and (X/-2,0). Making use of the

Tom M. Apostol