The exponential function possesses the properties
| (i.) | exp (x + y) = exp x × exp y. |
| (ii.) | (d/dx) exp x = exp x. |
| (iii.) | exp x = 1 + x + x2/2! + x3/3! + ... |
From (i.) and (ii.) it may be deduced that
exp x = (1 + 1 + 1/2! + 1/3! + ... )x,
where the right-hand side denotes the positive xth power of the number 1 + 1 + 1/2! + 1/3! + ... usually denoted by e. It is customary, therefore, to denote the exponential function by ex and the result
ex = 1 + x + x2/2! + x3/3! ...
is known as the exponential theorem.
The definitions of the logarithmic and exponential functions may be extended to complex values of x. Thus if x = ξ + iη
| log x = ∫x1 | dt |
| t |
where the path of integration in the plane of the complex variable t is any curve which does not pass through the origin; but now log x is not a uniform function, that is to say, if x describes a closed curve it does not follow that log x also describes a closed curve: in fact we have