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8.2. BRANCH CUTS 199
8.2 Branch Cuts
In the previous section, we observed that there was a difference between “expressions”,
whether concrete formulae or abstract expressions in K(x), and functions
R → R or C → C. If we go beyond K(x), the problems get more serious.
Example 13 Consider the formula sqrt(z), or √ z, which we can formalise
as θ ∈ K(z, θ|θ(z) 2 = z). What happens if we try to interpret θ as a function
C → C? Presumably we choose θ(1) = 1, and, by continuity, θ(1 + ɛ) =
1 + 1 2 ɛ − 1 8 ɛ2 + · · · ≈ 1 + 1 2 ɛ. Similarly, θ(1 + ɛi) ≈ 1 + 1 2ɛi, and so on. If we
continue to track θ(z) around the unit circle {z = x + iy : x 2 y 2 = 1}, we see
that θ(i) = 1+i √
2
, and θ(−1) = i. Continuing, θ(−i) = −1+i √
2
and θ(1) = −1.
It would be tempting to dismiss this as “the usual square root ambiguity”, but
in fact the problem is not so easily dismissed.
Example 14 Similarly, consider log(z), which we can formalise as θ ∈ K(z,
θ| exp(θ(z)) = z}. We chose θ(1) = 0, and, since exp(ɛi) ≈ ɛi, θ(ɛi) ≈ ɛi.
As we track θ(z) around the unit circle {z = x + iy : x 2 y 2 = 1}, we see that
θ(i) = iπ/2, θ(−1) = iπ, and ultimately θ(1) = 2πi.
The blunt facts are the following (and many analogues).
Proposition 56 (No square root function) There is no continuous function
f : C → C (or C \ {0} → C \ {0}) with the property that ∀z : f(z) 2 = z.
Proposition 57 (No logarithm function) There is no continuous function
f : C \ {0} → C \ {0} with the property that ∀z : f(exp(z)) = z.
In particular, the common statements
log z 1 + log z 2
?
= log z1 z 2 . (8.9)
log 1/z ? = − log z. (8.10)
are false for (any interpretation of) the function log : C \ {0} → C \ {0}, even
though they are true for the usual log : R + → R.
These facts have, of course, been known since the beginnings of complex
analysis, and there are various approaches to the problem, discussed in [Dav10].
Multivalued Functions Here we think of the logarithm of 1 as being, not
0, but any of {. . . , −4πi, −2πi, 0, 2πi, −4πi, . . .}. Our exemplar problem
(8.9) is then treated as follows.
The equation merely states that the sum of one of the (infinitely
many) logarithms of z 1 and one of the (infinitely many) logarithms
of z 2 can be found among the (infinitely many) logarithms
of z 1 z 2 , and conversely every logarithm of z 1 z 2 can be
represented as a sum of this kind (with a suitable choice of log z 1
and log z 2 ).
[Car58, pp. 259–260] (our notation)