228 
POWER SERIES 
184. The Hyperbolic Functions. Closely related with the cir 
cular functions are the hyperbolic functions. These are defined 
by the equations 
sinli x 
e x — e~ 
(1 
cosh x = 
tanli x = 
sedi x = 
Since 
e x + e~ 
2 
sinli X 
cosh x 
1 
cosh x 
x 
e x + e x 
, cosechrr = 
sinh 
= 1 + f! + fl + f! + 
/V* rJl /yi'3 
„ -i I fcC/ . 
e =1 -ll + 2l-3! + 
we have 
/V» />’5 
Sinh * = r! + 3I + ÌT + 
1 , a: 2 a; 4 . 
cosh a: = 1 + — + — + ••• 
valid for every x. From these equations we see at once 
sinh (—*)== — sinh x ; cosh ( — x) = cosh a\ 
sinh 0 = 0. 
ar* 
sinh a; = 1 + — + — + 
dx 
A 
dx 
sy% 7>0 /yiD 
cosh a? = — + — + — + 
1! 
2! 4! 
3 
3~! 
cosh 0 = 1. 
= cosh x. 
• = sinh x. 
(2 
(3 
(4 
(5 
(6 
Let us now look at the graph of these functions. Since sinh x, 
cosh x are continuous functions, their graph is a continuous curve. 
For x > 0, sinh a; > 0 since* each term in 3) is > 0. The relation 
4) shows that cosh x is positive for every x. 
If x' > x > 0, sinh x' > sinh a;, since each term in 3) is greater 
for F than for x. The same may be seen from 5).