up next prev ptail tail

## Article 16Expansion 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:

 $\begin{array}{ccc}\hfill tanhu& =u-\frac{1}{3}{u}^{3}+\frac{2}{15}{u}^{5}+\frac{17}{315}{u}^{7}+\dots ,\hfill & \hfill \\ \hfill sechu& =1-\frac{1}{2}{u}^{2}+\frac{5}{24}{u}^{4}-\frac{61}{720}{u}^{6}+\dots ,\hfill \\ \hfill ucothu& =1+\frac{1}{3}{u}^{2}-\frac{1}{45}{u}^{4}+\frac{2}{945}{u}^{6}-\dots ,\hfill \\ \hfill ucschu& =1-\frac{1}{6}{u}^{2}+\frac{7}{360}{u}^{4}-\frac{31}{15120}{u}^{6}+\dots .\hfill \end{array}}$ (29)

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.

 up next prev ptail top