302 Chapter 5
Similarly, we write w = cosh-^1 z to denote any value of w such that
cosh w == z. This equation is equivalent to cos iw = z, so
w = cosh-^1 z = -i arc cos z
= ± Log(z + v1z2="1) + 2krri
Here the principal branch is defined to be
w = Cosh-^1 z = Log(z + v1z2="1)
(5.25-10)(5.25-11)
To construct the lliemann surface for w = arc sin z we need to recall
the discussion of w = sinz in Section 5.19 and, in particular, to return to
Fig. 5.24, which shows geometrically the mapping defined by this function.
Since each strip
Sk = {z: %(2k - l)?T < Rez < %(2k + l)?T}
is mapped in a one-to-one manner onto
Cw - {(-oo, -1] U [+1, +oo)}and the boundary lines z = %(2lc + 1 )?T + iy are mapped twice on either
(-oo,-1] or [+l,+oo), depending on whether k is odd or even, in order
to obtain a one-to-one mapping between a conveniently enlarged strip and
a cut w-plane, we proceed as follows: Let
and
H2k = S2k U {z: z = 1/ 2 (4k-l)?T + y,y < O}
U{z: z=%(4k+l)?T+y,y~O}H2k+1 = S2k+1 U {z: z = %(4k + l)?T + y, y < O}
U{z: z=^1 M4k+3)?T+y,y~O}where k = 0, ±1, ±2,.... In Fig. 5.44 the boundaries of the H 2 k are
shown in heavy lines, while those of H 2 k+i are shown in thin lines. Then
we take thew-plane cut along the open intervals (-oo, -1) and ( +1, +oo ),
and for any given H 2 k we let the upper edge of [+l,+oo) correspond to
z = %(4k + l)?T + y, y ~ 0 and the lower edge of (-oo,-1) correspond to
z =^1 M 4k - l)?T + y, y < 0. As to each strip H 2 k+l we let the lower edge of
( +1, +oo) correspond to z = %( 4k + 1 )?T + y, y < 0, while the upper edge
of (-oo, -1] is made to correspond to z = %( 4k + 3)?T + y, y ~ 0.
Now considering the inverse function w = arc sin z, first we need to
interchange the roles of the z-and w-planes. Next, to build up the lliemann
surface for w = arcsinz, we take infinitely many copies Gj (j = O, ±1, ... )
of the cut z-planes, and each G j is put in a one-to-one correspondence