276 Chapter^5
1 1 '
sechz = cschz = --
cosh z ' sinh z
where coth z and csch z are not defined for z = krri, while tanh z, sech z
are not defined for z = %(2k + l)rri.
From (5.19-6) the following identities follow:- ez = coshz + sinhz, e-z = coshz - sinhz
- cosh(-z) ·= coshz, sinh(-z) = - sinh(z)
- cosh(z + 2krri) = coshz,sinh(z + 2k'ffi) = sinhz
- cosh^2 z - sinh^2 z = 1
- 1 - tanh^2 z = sech^2 z, coth^2 z - 1 = csch^2 z
6. cosh( z 1 + z 2 ) = cosh z 1 cosh z2 + sinh z1 sinh z2
7. sinh(z 1 + z 2 ) = sinhz 1 coshz 2 + coshz 1 sinhz 2
etc.
In the complex case the hyperbolic functions are not essentially different
from the circular functions. In fact, we have
and similarly,eiz _ e-iz
sinh iz = = i sin z
2
eiz + e-iz
cosh iz = ----= cos z
2tanhiz = itanz
etc.sin iz = i sinh zcos iz = cosh z
taniz = itanhz
etc.(5.19-7)(5.19-8)
Hence each circular identity implies a corresponding hyperbolic identity,and vice versa. For instance, cos^2 iz + sin^2 iz = 1 implies that cosh^2 z -
sinh^2 z == 1.Formulas (5.19-8) may be used to separate sin z, cos z, tan z, ... into
real and imaginary parts, and similarly, (5.19-7) may be used to the same
purpose in connection with hyperbolic functions. For instance, we have
w = u +iv= sin(x + iy) = sinxcosiy + cosxsiniy
= sin x cosh y + i cos x sinh y (5.19-9)
so that
u = sinx cosh y, v = cosxsinhy (5.19-10)