Page 25 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 2.4 Summary

A function  : I  C from a set I of reals into the complex numbers C is actually a familiar

concept from elementary calculus. It is simply a function from a subset of the reals into the
plane, what we sometimes call a vector-valued function. Assuming the function  is nice,
it provides a vector, or parametric, description of a curve. Thus, the set of all {(t) : (t) = e it
= cos t + i sin t = (cos t, sint), 0  t  2} is the circle of radius one, centered at the origin.

We also already know about the derivatives of such functions. If (t) = x(t) + iy(t), then the
derivative of  is simply ’(t) = x’(t) + iy’(t), interpreted as a vector in the plane, it is tangent
to the curve described by  at the point (t).
The real excitement begins when we consider function f : D  C in which the domain D is

a subset of the complex numbers. In some sense, these too are familiar to us from elementary
calculus—they are simply functions from a subset of the plane into the plane:
f(z) = f(x, y) = u(x, y) + iv(x, y) = (u(x, y), v(x, y))
Thus f(z) = z looks like f(z) = z = (x + iy) = x – y + 2xyi. In other words, u(x, y) = x – y 2
2
2
2
2
2
2
and v(x, y) = 2xy. The complex perspective, as we shall see, generally provides richer and
more profitable insights into these functions.
The definition of the limit of a function f at a point z = z is essentially the same as that
0
which we learned in elementary calculus:
lim f(z) L

z 0 z
2.5 Keywords
Elementary calculus: A function  : I  C from a set I of reals into the complex numbers C is
actually a familiar concept from elementary calculus.

Limit of a function: The definition of the limit of a function f at a point z = z is essentially the
0
same as that which we learned in elementary calculus.
Derivatives: Suppose the function f given by f(z) = u(x, y) + iv(x, y) has a derivative at z = z =
0
(x , y ). We know this means there is a number f’(z ) so that
0
0
0

0
0
f'(z )  lim f(z   z) f(z )
0
 z 0  z
2.6 Self Assessment
1. A function  : I  C from a set I of reals into the complex numbers C is actually a familiar
concept from ......................
2. The real excitement begins when we consider function ...................... in which the domain
D is a subset of the complex numbers.
3. The definition of the ...................... f at a point z = z is essentially the same as that which we
0
learned in elementary calculus.
4. If f has a derivative at z , we say that f is ...................... at z .
0
0
5. Suppose the function f given by f(z) = u(x, y) + iv(x, y) has a derivative at z = z = (x , y ). We
0
0
0
know this means there is a number f’(z ) so that ......................
0

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