COMPLEX ANALYSIS: Maple Worksheets, 2001
(c) John H. Mathews Russell W. Howell
mathews@fullerton.edu howell@westmont.edu

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COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9
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CHAPTER 2 COMPLEX FUNCTIONS

Section 2.5 Branches of Functions

In Section 2.3 we defined the principal square root function and investigated some of its properties. We left some unanswered questions concerning the choices of square roots. We now look into this problem because it is similar to situations involving other elementary functions.

In our definition of a function in Section 2.1 we specified that each value of the independent variable in the domain is mapped onto one and only one value of the dependent variable. As a result, one often talks about a single-valued function , which emphasizes the only one part of the definition and allows us to distinguish such functions from multiple-valued functions, which we now introduce.

Let denote a function whose domain is the set and whose range is the set . If is a value in the range, then there is an associated inverse relation that assigns to each value the value (or values) of in for which the equation holds true. But unless takes on the value at most once in , then the inverse relation is necessarily many valued, and we say that is a multivalued function . For example, the inverse of the function is the square root function . We see that for each value other than , the two points and are mapped onto the same point ; hence is in general a two-valued function.

The study of limits, continuity, and derivatives loses all meaning if an arbitrary or ambiguous assignment of function values is made. For this reason we did not allow multivalued functions to be considered when we defined these concepts. When working with inverse functions, it is necessary to carefully specify one of the many possible inverse values when constructing an inverse function. The idea is the same as determining implicit functions in calculus. If the values of a function are determined by an equation that they satisfy rather than by an explicit formula, then we say that the function is defined implicitly or that is an implicit function . In the theory of complex variables we study a similar concept.

Let be a multiple-valued function. A branch of is any single-valued function that is continuous in some domain (except, perhaps, on the boundary), and at each point in the domain, assigns one of the values of .

Load Maple's "conformal mapping" procedure.
Make sure this is done only ONCE during a Maple session.

> with(plots):

Warning, the name changecoords has been redefined

Example 2.20, Page 79. Consider the two branches of the square root function:
= , and
= .

(a) Find the image of the disk in the -plane slit along
the ray , under the mapping .

> f1:='f1': z:='z':
f1 := z -> z^(1/2):
`f1(z) ` = f1(z);
conformal(f1(Re(z)*exp(I*Im(z))), z=0.01-I*3.14..4+I*3.14,
title=`w = f1(z) = z^(1/2)`,
grid=[13,13], numxy=[13,13],
scaling=constrained,
view=[-2..2,-2..2]);

(b) Find the image of the disk in the z-plane slit along
the ray , under the mapping .

> f2:='f2': z:='z':
f2 := z -> - z^(1/2):
`f2(z) ` = f2(z);
conformal(f2(Re(z)*exp(I*Im(z))), z=0.01-I*3.14..4+I*3.14,
title=`w = f2(z) = - z^(1/2)`,
grid=[13,13], numxy=[13,13],
scaling=constrained,
view=[-2..2,-2..2]);

End of Section 2.5.