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+
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