DIFFERENTIAL EQUATIONS 
337 
whose roots are m y = 2, m. 2 = 3. Hence two solutions of (3) are e 2x 
and e 3x . These are evidently linearly independent, and so the general 
solution of (3) is 
y — Ae* + Be 3x . 
Imaginary Roots of (2). Suppose, however, that (2) has no real 
roots, i.e. suppose that, on writing down the formal solution of (2), 
m = — a ± V« 2 — /3, 
it turns out that a 2 — /3 < 0. The roots of (2) are then imaginary, 
as the mathematicians of the eighteenth century said. They can be 
expressed in the form 
(4) m x = — a + yV — 1, m 2 = — a — y V— 1, 
where y=V/? — a 2 . The mathematicians of that time did not 
hesitate to work with imaginary expressions like the above, even 
though they had no clear idea of what they mean, i.e. how to define 
them. They reasoned as follows. Since e“ 4 ’’ = e u e v when u and v 
are real, the expression 
must be the same thing as the product 
g-oxg-yxvCl 
and so the question reduces itself to that of finding out what 
means, where <j> is a real number. 
Now, the mathematicians of that time were very well acquainted 
with the expansions of the functions e x , since, coscc by Taylor’s 
Theorem : 
=1+X+—+—+ 
2:3! 
• cc 3 . æ 5 
Sin X = X 
3! 5! 
What could be more natural, therefore, than to ask the series what 
e**- 1 means? On setting cc=(/>V—1 in the above development 
of e x and reducing the result by means of the relations 
(■v=ï) 1 = V=ï, ( V=1 ) 2 = -1, ( v^l) 3 = V=I, 
( v^l) 4 = l, (V^l) 4t+l = (V^iy, 
= 1, 2, ... ; *«1,2, 3,4,