1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 291

00 00

-- - -1---......

0

Fig. 5.33

z-planes (or those of the corresponding spheres) are to be made, consider
small circles C 1 , C 2 about the points zi, z 2 , respectively, as well as a circle
Ca enclosing both points and a circle C 4 without those points (Fig. 5.34).
Since

arg w = % arg a + % arg( z - z1) + % arg( z - z2)

it is clear that if z describes either C 1 or C 2 once in the positive direction,
then argw increases by 71', thus leading from a value w 0 in one branch of
the function to the value w 1 = -w 0 in the other branch of that function.
However, if z describes Ca once in the positive direction, then arg w in-
creases by 271', thus restoring the starting value of w, while in a description
around C 4 , arg w will come back to its initial value. That is, in the last
two cases the branches of w are not interchanged. Therefore, to construct
the Riemann surface of the given function it suffices to cut two copies of
the z-plane along a line from z1 to z 2 , put one on top of the other, and
connect crosswise the edges of the cuts.

y

0

0 x

Fig. 5.34