Article 16
Expansion of Hyperbolic Functions.

For this purpose take Maclaurin’s Theorem,

and put

then

hence

and similarly, or by differentiation,

By means of these series the numerical values of $sinhu,coshu$, can be computed and tabulated for successive values of the independent variable $u$. They are convergent for all values of $u$, because the ratio of the $n$th term to the preceding is in the first case , and in the second case , both of which ratios can be made less than unity by taking $n$ large enough, no matter what value $u$ has. Lagrange’s remainder shows equivalence of function and series.

From these series the following can be obtained by division:

These four developments are seldom used, as there is no observable law in the coefficients, and as the functions $tanhu,sechu,cothu,cschu$, can be found directly from the previously computed values of $coshu,sinhu$.

Prob. 36.

Show that these six developments can be adapted to the circular functions by changing the alternate signs.