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202 CHAPTER 8. ALGEBRA VERSUS ANALYSIS
• A rule for computing f(z) on B, which is generally arranged to make
f(z) continuous with one side or the other of B.
A typical example would be log, where we might have:
• Initial value (1, 0);
• Branch cut {x + 0i : x ∈ (−∞, 0]};
• Rule for log x : x < 0: log(x + 0i) = lim ɛ→0 + log(x + ɛi).
We should note that there are many possible choices: the author was in
fact initially taught
• Initial value (1, 0);
• Branch cut {x + 0i : x ∈ [0, ∞)};
• Rule for log x : x > 0: log(x + 0i) = lim ɛ→0 + log(x + ɛi).
Despite the apparent arbitrariness, the world has tended to converge on
the branch cuts as defined 3 in [AS64].
8.2.1 Removable Branch Cuts
This term is introduced in [?, §4.2], presumably by analogy with “removable
signularity”.
8.3 Fundamental Theorem of Calculus Revisited
In the previous chapter we reduced the Fundamental Theorem of Calculus to
the status of Notation 22.
8.4 Constants Revisited
In the previous chapter we defined (Definition 81) a constant as being an element
whose derivative was zero. How well does this compare with our usual intuition,
which is of a constant function, i.e. one whose value is independent of the
argument?
It is a truism of calculus that a constant function has to have derivative zero.
8.4.1 Constants can be useful
We can make constructive use of the concept of a differential constant, however.
Mathematica, for example, writes [Lic11] √ [ ]
x 2 in the form x , and then
treats [· · ·] as a differential constant.
3 The careful reader should note that one should use printing 9 or later: the branch cut for
arccot moved!
x 2
x