526 
FUNCTION. 
[787 
4. The foregoing equation exp (x + y) = exp x. exp y is, in fact, the equation of 
indices, a x+y = a x .a y ; exp x is thus the same thing as (?, where e denotes a properly- 
determined number, and putting e x equal to the series, and then writing x—1, we 
have e=l + \ + ^s + &c., that is, e = 2'7128 ... But as well theoretically as 
1 1 . Z 1 • Z . o 
for convenience of printing, there is considerable advantage in the use of the notation 
exp u. 
From the equation, exp iy = cos y + i sin y, we deduce exp (— iy) = cos y — i sin y, and 
thence 
cos y = {exp (iy) + exp (- iy)}, 
sin V = ^ {exp (iy) - exp (- iy)}; 
if we write herein ix instead of y we have 
cos ix = \ {exp x + exp ( — x)}, 
sin ix = ^ {exp x — exp (— x)}, 
viz. these values are 
xr 
COS IX = 1 + —- + 
1 . z 
1 . . X? 
- sin ix = x + ■—a _ 
i 1.2.3 
x 4 
1.2.3.4 
+ ... 
+ ... 
each of them real when x is real. 
The functions in question 
1 + g + 
ÎC 4 
27374 + '" 
and x + - g 3 +..., regarded as functions of are termed the hyperbolic cosine and 
sine, and are represented by the notations cosh x and sinh x respectively; and similarly 
we have the hyperbolic tangent tanh x, &c.: although it is easy to remember that 
cos ix, sin ix, are, in fact, real functions of x, and to understand accordingly the formulae 
wherein they occur, yet the use of these notations of the hyperbolic functions is often 
convenient. 
5. Writing u — exp v, then v is conversely a function of u which is called the 
logarithm (hyperbolic logarithm, to distinguish it from the tabular or Briggian logarithm), 
and we write v = log u, or what is the same thing, we have u = exp (log u): and it is 
clear that if u be real and positive there is always a real and positive value of logu, 
in particular the real logarithm of e is = 1; it is however to be observed that the 
logarithm is not a one-valued function, but has an infinity of values corresponding to 
the different integer values of a constant s; in fact, if logw be any one of its values, 
then • log u + 2s7n is also a value, for we have exp (log u + 2sTri) = exp log u exp 2s7ri, or 
since exp 2sTri is =1, this is =u; that is, \ogu+2sTri is a value of the logarithm of u. 
We have 
uv = exp (log uv) = exp log u. exp log v,