Elementary Functions 299contrast, the Riemann surface of w = * ::fZ is closed: i.e., its branch points
belong to the surface.5.24 The General Power FUnction
If a is a given complex constant, the function w = z,,, defined on C-{O} by
(5.24-1)
is called the general power function. From the discussion in Section 1.16
it follows that this function is single-valued iff a = n (n a nonnegative
integer). When a = 1/n (n a positive integer greater than 1) and the
definition is completed with 01 /n = 0, the function coincides with the nth
root w = * ::/Z, so it is multiple-valued with a finite number of values. More
generally, w = zm/n = *\l'Z"', with m, n integers; gcd(m,n) = 1, n ~ 2,
is also many valued. However, for a irrational or imaginary the function
has infinitely many values.
In the multiple-valued case the principal value or principal branch of
the function is defined by
(5.24-2)
Suppose that a is not a rational number, and let z describe a circle
about the origin, once in the positive direction. Then a given value of
log z will continuously change into log z + 27ri when z returns to its original
position. As a consequence, the corresponding value of z,,, will become
zOle^2 "ioi. After·.a second turn about the origin, in the same direction, z,,,
becomes z,,,e^4 "ioi, and so on. On the other hand, if z describes the circle
in the negative direction, after one circuit the function will assume the
value zOle-^2 "ia, after two circuits the value zae-^4 "ia, and so on. Hence
any possible value of z,,, can be reached in this manner, so the Riemann
surface for zOI is similar to that for log z.
If a is an irrational real number, we have lz,,,e^2 k7rioil = lz,,,I, which shows
that all images of the same z will have the same modulus; i.e., they will lie
on the same circle on the w-plane, their arguments differing by multiples of
27ra. However, if a = (J + i-y, with 'Y -=f. O, and we let z = reiO, we find that
W = e<.B+ir)[in r+i(o+2k7r)]
= e.Bln r-")'(9+2k7r) ei['Yln r+.8(o+2k7r))so that
p = lw I = r.B e-'Y( o+2k7r)
and all images of the same z will lie on a logarithmic spiral.