194 Functions of a Complex Variable
imply
ur = r~^1 ve, vr = -r~^1 ue, (4.2.3)
and conversely, (4-2.3) imply (4-2.2). (b) Verify that each of the following
functions satisfies (4-2.3): (i) f(z) = raelaS, a a real number; (ii) f{z) =
lnr + i6
The usual rules of differentiation (for the sum, difference, product, ratio,
and composition (chain rule) of two functions, and for the inverse of a
function) hold for the functions of complex variable just as for the functions
of real variable. If the function / = f(x) is differentiable, the chances
are good that the corresponding function / = f(z) is analytic, and to
compute the derivative of f(z), you treat z the same way as you would
treat x. The functions f{z) that contain z, $tz, $Sz, arg(z), and \z\ require
special attention because they do not have clear analogs in the real domain.
Analyticity of such functions must be studied using the Cauchy-Rieman
equations. FOR EXAMPLE, the function f(z) = z is not analytic anywhere,
because for this function ux = 1, vy = — 1, and so ux ^ vy.
EXERCISE 4.2.11.c Check whether the following functions f are analytic.
If the function is analytic, find the derivative, (a) f(z) = (z)^2 , (b) f(z) =
3z/3fcz, (c) f{z) = 8fcz^3 - iSz^3 , (d) f(z) = z^2 /(l - z^2 ).
4.2.3 The Integral Theorem and Formula of Cauchy
In this section, we study integrals of analytic functions and establish two
results from which many of the properties of the analytic functions follow.
We start with integration in the complex plain. Consider a curve C in
R^2 denned by the vector-valued function r(t) = x(t)i + y(t)j, a < t <
b. Equivalently, we can define this curve using a complex-valued function
z = z(t), t £ [a, b], by setting z(t) ~ x(t) + iy(t). As usual, we write
z(t) = x(t) + iy(t), provided the derivatives of x and y exist.
Assume that the curve C is piece-wise smooth, that is, consists of finitely
many smooth pieces; see page 28 for details. Let / = f(z) be a function,
continuous in some domain containing the curve C. We define the integral
Jc f(z)dz of / along C by
/ f(z)dz = f f(z(t))z(t)dt. (4.2.4)
JC Ja
In what follows, we consider only piece-wise smooth curves. The line inte-