320] 
ON A TRANSCENDENT EQUATION. 
87 
or if 
then 
_ h ie e* u + e-* u + i (<?£“ - e~* u ) 
£ gl'tt g—i _ l (gi" _ g<~hu) 
__, ^ é" + 1 +i (e u — 1) 
- e u +l-i(e u -1) 
, l+itan^d) , ., 
= l0g l-»tan 1 0 = |O S^ = ^. 
U = log tan(|7T + £<£), 
(f> = 1 log tan (¿7T + §wi) ; 
and substituting for </> its value, we obtain 
gd u = ^ log tan (j 7t + %ui), 
which is the definition of the transcendent gd u. It is to be noticed that gd u, 
although exhibited in an imaginary form, is a real function of u; and, moreover, that 
it is an odd function, viz. we have 
and therefore also 
The original equation, 
written under the form 
shows that we have 
gd (- u) = - gd (u), 
gd (0) = 0. 
u= log tan (| 7T+ £</>), 
u = i \ log tan ^ 7t + 1. i v j 
u = i gd (i ) = i gd (-itf>); 
or substituting for </> its value gd a, we have 
u = i gd (— i gd it), 
which may also be written 
iu = gd (i gd it); 
so that gd u is a quasi-periodic function of the second order—a property which has 
not, at least obviously, any analogue in the general theory. We have 
cos gd u = \ (e' gd " + e -tgd “) 
l /1 + tan \ui 1 — tan ^ui\ 
~~ 2 Vl — tan \ id 1 + tan \u%] 
1 + tan- i ni _ 1 
1 — tan 2 \ ui cos ui’