Chapter 0
Differentiability and
the Cauchy-Riemann
Equations
A few preliminaries are in order. The letter C will denote the set of all
complex numbers.
This raises the question of exactly what a complex number is. In most
places in this book (and indeed in most situations in mathematics) one thinks
of a complex number as "a quantity of the form x + iy, where x, y G R and
i2 = —1"; one thinks of the ordered pair (x,y) as a way of "representing"
the complex number x + iy as a point in the plane. The present chapter
is one of the few places where we need to keep in mind that in fact x + iy
is literally equal to (x,y), by definition. (There are many possible ways to
define the complex numbers, but saying that x + iy (x, y) is the simplest
and most common; if the idea that x + iy (x, y) is not familiar then you
should read Appendix 1 before proceeding further.)
Now, since C =
R2,
so in particular C is a real vector space as well as
being a field in its own right, there are at least two things that one might
mean by saying that T : C —- C is "linear". Recall that T is linear if for
every z, w £ C and every scalar c we have
T(z +
w)
=
Tz
+ Tw,
T(cz)
= cTz.
If this holds for all real scalars c we will say that T is R- linear, while if
it holds for all complex scalars c we will say that T is C- linear.
~ 3
http://dx.doi.org/10.1090/gsm/097/01
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