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Two Mathematical Identities

This page discusses two mathematical equations. Both of these equations were discovered by the great mathematician Leonhard Euler, whose extensive mathematical publications have served as the foundation for much of present-day mathematics. One has been presented to nonmathematicians as something profound, but is perhaps not quite as profound as claimed, and the other is more genuinely beautiful, but has been known mostly only to mathematicians.

The first one is:

  i * pi
e        = -1

and the second one is:

               1      1      1      1
zeta(x) = 1 + ---- + ---- + ---- + ---- + ... =
                x      x      x      x
               2      3      4      5

     1              1              1              1
------------ * ------------ * ------------ * ------------ * ...
       1              1              1              1
( 1 - ---- )   ( 1 - ---- )   ( 1 - ---- )   ( 1 - ---- )
        x              x              x              x
       2              3              5              7

which is true when (the real part of x) is bigger than 1.



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The first one is:

e = 1

and the second one is:

ζ ( x ) = 1 + 1 2 x + 1 3 x + 1 4 x + 1 5 x + . . . = i = 1 1 i x

= 1 ( 1 - 1 2 x ) × 1 ( 1 - 1 3 x ) × 1 ( 1 - 1 5 x ) × 1 ( 1 - 1 7 x ) × ... = p P 1 ( 1 - 1 p x )

which is true when (the real part of) x is greater than 1.