Plane trigonometry, by S.L. Loney.

HYPERBOLIC FUNCTIONS.

373

317. From the values in Art. 313 it follows, by
Art. 302, that
cosh x = (ex + e-~)
X2 X4 X6
=1+ 2 + F +......
sinh x = - [ez - e-x]
X3 X5 X7
= +3- - 5+- +.......
These are the expansional values of cosh x and sinh x.
#318. Periods of the hyperbolic functions.
For all values of 0, real or complex, we have cos Oi = cosh 0.
Hence
cosh(x + yi) =cos {(x +yi) i} = cos (xi-y) =cos [ - 2r+xi - y] (Art. 311)
=cos [(27ri + x + yi) i] = cosh [2wri + x +yi]
= (similarly) cosh [47ri + x + yi] =......
Hence the hyperbolic cosine is periodic, its period being imaginary
and equal to 27ri.
Again, since sinh 0 = - i sin Oi, we have
sinh (x + yi) - i sin (x + yi) i} = - i sin [xi - y]
= -i sin [ - 27r + xi - y] = -isin {[27ri+ x+yi]i}
= sinh [27ri + x + yi],
so that the period of sinh (x+yi) is 27ri.
Similarly it may be shewn that the period of tanh (x +yi) is tri.
The hyperbolic functions therefore differ from the circular functions
in having no real period; their period is imaginary.
319. Ex. 1. Separate into its real and imaginary parts the expression
sin (a + pi).