172 CHAPTER 5 • ELEMENTARY FUNCTIONSIdentity (5-22) yields the principal values of JI+i and ii:
J1+i=(l+i)~=exp [~Log(l + i)]
=exp[~ ( 1n2! +i~)](
=exp Ln2•^1 + i.'Tr)8
= 21 • ( cos7r.. 'Tr)8
+ism
8
::::: 1.09684 + 0.45509i and
ii = exp (iLog < i) I= exp [i (ii))
= exp (-i)
::::: 0.20788.Note that the result of raising a complex number to a complex power may
be a real number in a nontrivial way.We now consider the possibilities that arise when we apply Equation (5-21).Case (i) Suppose that c = k, where k is an integer. Then, if z = rei^9 =ft 0,
klog(z) = {kln(r) +ik(8+ 2n7r): n is an integer}.Recalling that the complex exponential function has period 27ri, we havezk = exp [k log (z)]
= exp [k In (r) + ik (8 + 2n7r)]
= exp [In (rk) + ikB + i2kn7r]
= exp (in (r")] exp (ik8) exp (i2knir)
= rk exp ( ik9) = r" (cos kB + i sin kB),which is the single-valued kth power of z that we discussed in Section 1.5.
Case (ii): If c = ~.where k is an integer, and z = re^19 =ft 0, then
1 {1 i(B+ 2nir). }
k log z = k In r + k : n 1s an integer.