1. Establish the following addition formulae:
(a) cos(z + w) = cosz cosw – sin z sin w,
(b) sin(z+w) =sinzcosw+coszsinw,
(c) cosh(z + w) = cosh z cosh w + sinh z sinh w,
(d) sinh(z + w) = sinh z cosh w + cosh z sinh w,
2. Show that Icos zl2 = cos 2X + sinh2 y, where z = x + iy. Find all
zeros and periods of cos z.
3. Find all zeros and periods of cosh z and sinh z.
4. Show that
1 (1 +iZ)tan- 1 z = 2i log 1- iz ‘
where both sides of the identity are to be interpreted as subsets of
the complex plane. In other words, show that tan w = z if and only
if 2iw is one of the values of the logarithm featured on the right.
5. Let S denote the two slits along the imaginary axis in the complex
plane, one running from i to +ioo, the other from -i to -ioo. Show
that (1 + iz)/(l – iz) lies on the negative real axis (-00,0] if and 32 I The Complex Plane and Elementary Functions
only if z E S. Show that the principal branch
1 (1 + iZ)Tan- 1z = 2i Log 1- iz
maps the slit plane C\S one-to-one onto the vertical strip {I Re wi <
11’/2}.
6. Describe the Riemann surface for tan- 1 z.
7. Set w= cos z and ( = e iz . Show that ( = w± .Jw 2 – 1. Show that
cos- 1 w = -ilog [w ± “;w 2 -1] ,
where both sides of the identity are to be interpreted as subsets of
the complex plane.
8. Show that the vertical strip IRe(w) I < 11’/2 is mapped by the func-
tion z(w) = sinw one-to-one onto the complex z-plane with two
slits (-00, -1 J and [+1, +(0) on the real axis. Show that the in-
verse function is the branch of sin-1 z = -i Log (iz + .J1 – z2)
obtained by taking the principal value of the square root. Hint.
First show that the function 1 – Z2 on the slit plane omits the nega-
tive real axis, so that the principal value of the square root is defined
and continuous on the slit plane, with argument in the open interval
between -11’/2 and 11’/2
