the series, however, is convergent for real values of x only when x lies between +1 and -1. Other formulae which are deducible from this equation are given in the portion of this article relating to the calculation of logarithms. The function log x as x increases from 0 towards [oo] steadily increases from -[oo] towards +[oo]. It has the important property that it tends to infinity with x, but more slowly than any power of x, i.e. that x^{-m} log x tends to zero as x tends to [oo] for every positive value of m however small. The _exponential function_, exp x, may be defined as the inverse of the logarithm: thus x = exp y if y = log x. It is positive for all values of y and increases steadily from 0 toward [oo] as y increases from -[oo] towards +[oo]. As y tends towards [oo], exp y tends towards [oo] more rapidly than any power of y. The exponential function possesses the properties (i.) exp (x + y) = exp x × exp y. d (ii.) --- exp x = exp x. dx (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 e^x and the result e^x = 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 = [xi] + i[eta] _ / x dt log x = | --- _/ 1 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 log ([xi] + i[eta]) = log [root]([xi]² + [eta]²) + i([alpha] + 2n[pi]), where [alpha] is the numerically least angle whose cosine and sine are [xi]/[root]([xi]² + [eta]²) and [eta]/[root]([xi]² + [eta]²), and n denotes any integer. Thus even when the argument is real log x has an infinite number of values; for putting [eta] = 0 and taking [xi] positive, in which case [alpha] = 0, we obtain for log [xi] the infinite system of values log [xi] + 2n[pi]i. It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation log(1 + x) = x - ½x^2 + {1/3}x^3 - ¼x^4 + ...