5.4 • TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 187(h) cosh if.
(i) cosh (^4 ~'") •
7. Find the derivatives of the following, and state where they are defined.(a) sin(~).
(b) z tan z.
(c) secz^2 •
(d) z csc^2 z.
(e) z sinh z.
(f) coshz^2.
(g) z tan z.- Show that
(a) sin.Z = sinz holds for all z.
(b) sin z is nowhere analytic.
(c) cosh:Z = coshz holds for all z.
( d) cosh z is nowhere analytic.- Show that
(a) lim coo z- 1 = o.
z-0 z
(b) Jim tan ( xo + iy) = i, where xo is any fixed real number.
Y-+oo- Find all values of z for which each equation holds.
(a) sinz = cosh4.
(b) cosz = 2.
(c) sinz = isinhl.
(d) sinhz = ~·
(e) coshz = 1.1 1. Show that the zeros of sinz are at z = mr, where n is an integer.
12. Use Equation (5-36) to show that, for z = x + iy, lsinh YI ~ !sin z l ~ cosh y.- Use Identities (5-36) and (5-37) to help establish the inequality !cos zl^2 + lsin z l^2 ;:::
1, and show that equality holds iff z is a real number. - Show that the mapping w = sin z
(a) maps the y-axis one-to-one and onto the v-axis.
(b) maps the ray { (x, y) : x = f, y > 0} one-to-one and onto t he ray
{(u,v): u > 1, v = O}.