Complex Analysis - Maths KU

284 Chapter 5 Integration in the Complex Plane
5.6 Applications
Since the maximum modulus of f occurs on |z| = 1, the inequality shows that
the maximum modulus of f(z) =z 2 +5z − 1 over the region is 7.
33. In this problem we will start you out in the proof of the minimum modulus
theorem.
If f is analytic on a closed region R bounded by a simple closed curve
C and f(z) �= 0for all z in R, then the modulus |f(z)| attains its
minimum on C.
Define the function g(z) =1/f(z), reread Theorem 5.16, and then complete
the proof of the theorem.
34. Suppose f(z) =z +1−i is defined over the triangular region R that has vertices
i, 1, and 1 + i. Discuss how the concept of distance from the point −1+i can
be used to find the points on the boundary of R for which |f(z)| attains its
maximum value and its minimum value.
In Section 1.2 we 5.6introduced
the notion that a complex number could be interpreted as a
two-dimensional vector. Because of this, we saw in Section 2.7 that a two-dimensional vector
field F(x, y) =P (x, y)i + Q(x, y)j could be represented by means of a complex-valued
function f by taking the components P and Q of F to be the real and imaginary parts of
f; that is, f(z) =P (x, y) +iQ(x, y) is a vector whose initial point is z. In this section
we will explore the use of this complex representation of the vector function F(x, y) in the
context of analyzing certain aspects of fluid flow. Because the vector field consists of vectors
representing velocities at various points in the flow, F(x, y) orf(z) is called a velocity
field. The magnitude �F� of F , or the modulus |f(z)| of the complex representation f, is
called speed.
It will assumed throughout this section that every domain D is simply connected.
Irrotational Vector Field Throughout this section we consider
only a two-dimensional or planar flow of a fluid (see Sections 2.7 and 3.4).
This assumption permits us to analyze a single sheet of fluid flowing across
a domain D in the plane. Suppose that F(x, y) = P (x, y)i + Q(x, y)j
represents a steady-state velocity field of this fluid flow in D. In other words,
the velocity of the fluid at a point in the sheet depends only on its position
(x, y) and not on time t. In the study of fluids, if curl F = 0, then the fluid flow
is said to be irrotational. If a paddle device, such as shown in Figure 5.48,
is inserted in a flowing fluid, then the curl of its velocity field F is a measure
of the tendency of the fluid to turn the device about its vertical axis (imagine
this vertical axis pointing straight out of the page). The flows illustrated in
Figures 5.48(a) and 5.48(b) are irrotational because the paddle device is not
turning. The word “irrotational” is somewhat misleading because, as is seen
in Figure 5.48(b), it does not mean that the fluid does not rotate. Rather,
if curl F = 0, then the flow of the fluid is free of turbulence in the formof
vortices or whirlpools that would cause the paddle to turn. In the case of