168 CHAPTER 5 • ELEMENTARY FUNCTIONS
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Figure 5.5 T he Riemann surface for mapping w =log (z).
The Riemann surface for the multivalued function w = log (z) is similar to
the one we presented for the square root function. However, it requires infinitely
many copies of the z plane cut along the negative x-axis, which we label Sk for
k = ... , - n, ... , -1, 0, 1, ... , n, .... Now, we stack these cut planes directly
on each other so that the corresponding points have the same position. We
join the sheet Sk to Sk+i as follows. For each integer k , the edge of the sheet
sk in the upper half-plane is joined to the edge of the sheet sk+l in the lower
half-plane. The Riemann surface for the domain of log looks like a spiral stair-
case that extends upward on the sheets Si. S2,... and downward on the sheets
S 1 , S 2 , ... , as shown in Figure 5.5. We use polar coordinates for z on each
sheet. For Sk, we use
z = r (cose + isin8), where
r = lzl and 27rk - 7r < e $ 7r + 21rk.
Again, for Sk, the correct branch of log (z) on each sheet islog(z) = lnr+iB, where
r = lzl and 27rk - 7r < e $ 7r + 27rk.-------~EXERCISES FOR SECTION 5.2- Find all values for
{a) Log (ie^2 ).
{b} Log ( J3 -i).
(c) Log (iv'2 - v'2).
(d) Log [(1 + i)^4 ].