With it s emphasis on concepts and structures, moder n mathematics has mostly

shied away from formulae . Nevertheless , many of the greatest peak s of mathematic s

are described b y formulae o f one sort or another. Her e is a sample of some of them

that Eule r discovered .

elx

= cos x + i sin x

ein = -1

log(-l) = m + 2kiir(k = 0, ±1, ±2,...)

~ - ~ - ~

4 : r

2

n h -2TT2J i ««=I I *

X J " A V n J-J ; V (2fc+l)

27T2

n=l x / n— 1 x v 7

_1_ \_ J _ _ (-l)

fc~1 22fc-1

.B2fc

2f c

2fc +

3

f c +

' "

+

n

2fc

+ " • - (2fc) ! *

7T 1 x-^ , . _ / 1 1

S n= l V

n -+ - 5 n — 5

i ^ / i i

7T CO t S7T = h

5

z

—' V n + s n — 5

n=l

x

7T _ 1 1 1 1 1 1

7T 1 1 1 1 1

= 1 + " - - -- + - + -7 " .. •

2^2 3 5 7 9 1 1

7T _ 1 1 1 1 1

3 " + 5 ~ 7~ 11+ 13 + 1 7 ' "

7 T 2 _ J L _ _ J _ _1 _ ^ _

8^2 3

2

5

2

7

2

9

2

7T 2 1 1 1 1

1 - -=* ~ ^ + 7

6 ^ 5 2 7 2 l

l7^2 132+