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.

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http://dx.doi.org/10.1090/gsm/097/01