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 + ...
is true only when the analytical modulus of x is less than unity. The
exponential function, which may still be defined as the inverse of the
logarithmic function, is, on the other hand, a uniform function of x,
and its fundamental properties may be stated in the same form as for
real values of x. Also
exp ([xi] - i[eta]) = e^{[xi]}(cos [eta] + i sin [eta]).
An alternative method of developing the theory of the exponential
function is to start from the definition
exp x = 1 + x + x²/2! + x³/3! + ...,
the series on the right-hand being convergent for all values of x and
therefore defining an analytical function of x which is uniform and
regular all over the plane.
_Invention and Early History of Logarithms._--The invention of
logarithms has been accorded to John Napier, baron of Merchiston in
Scotland, with a unanimity which is rare with regard to important
scientific discoveries: in fact, with the exception of the tables of
Justus Byrgius, which will be referred to further on, there seems to
have been no other mathematician of the time whose mind had conceived
the principle on which logarithms depend, and no partial anticipations
of the discovery are met with in previous writers.
The first announcement of the invention was made in Napier's _Mirifici
Logarithmorum Canonis Descriptio ..._ (Edinburgh, 1614). The work is a
small quarto containing fifty-seven pages of explanatory matter and a
table of ninety pages (see NAPIER, JOHN). The nature of logarithms is
explained by reference to the motion of points in a straight line, and
the principle upon which they are based is that of the correspondence of
a geometrical and an arithmetical series of numbers. The table gives the
logarithms of sines for every minute of seven figures; it is arranged
semi-quadrantally, so that the _differentiae_, which are the differences
of the two logarithms in the same line, are the logarithms of the
tangents. Napier's logarithms are not the logarithms now termed
Napierian or hyperbolic, that is to say, logarithms to the base e where
e = 2.7182818...; the relation between N (a sine) and L its logarithm,
as defined in the _Canonis Descriptio_, being N = 10^7e^{-L/(l0^7)}, so
that (ignoring the factors 10^7, the effect of which is to render sines
and logarithms integral to 7 figures), the base is e^{-l}. Napier's
logarithms decrease as the sines increase. If l denotes the logarithm to
base e (that is, the so-called "Napierian" or hyperbolic logarithm) and
L denotes, as above, "Napier's" logarithm, the connexion between l and L
is expressed by