FUNCTIONS OF A COMPLEX VARIABLE in .NET

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FUNCTIONS OF A COMPLEX VARIABLE
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derivatives at points within C can be calculated. Thus, if a function of a complex variable has a rst derivative, it has all higher derivatives as well. This, of course, is not necessarily true for functions of real variables.
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TAYLOR S SERIES Let f z be analytic inside and on a circle having its center at z a. Then for all points z in the circle we have the Taylor series representation of f z given by f z f a f 0 a z a See Problem 16.21. f 00 a f 000 a z a 2 z a 3 2! 3! 8
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SINGULAR POINTS A singular point of a function f z is a value of z at which f z fails to be analytic. If f z is analytic everywhere in some region except at an interior point z a, we call z a an isolated singularity of f z .
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EXAMPLE. If f z 1 , then z 3 is an isolated singularity of f z . z 3 2
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sin z has a singularity at z 0. Because lim is nite, this singularity is called a EXAMPLE. The function f z z!0 z removable singularity.
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POLES
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 z ;  a 6 0, where  z is analytic everywhere in a region including z a, and if n is a z a n positive integer, then f z has an isolated singularity at z a, which is called a pole of order n. If n 1, the pole is often called a simple pole; if n 2, it is called a double pole, and so on. If f z
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LAURENT S SERIES If f z has a pole of order n at z a but is analytic at every other point inside and on a circle C with center at a, then z a n f z is analytic at all points inside and on C and has a Taylor series about z a so that a n a n 1 a 1 a0 a1 z a a2 z a 2 9 f z z a n z a n 1 z a This is called a Laurent series for f z . The part a0 a1 z a a2 z a 2 is called the analytic part, while the remainder consisting of inverse powers of z a is called the principal part. More 1 X ak z a k as a Laurent series, where the terms with k < 0 constitute generally, we refer to the series
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the principal part. A function which is analytic in a region bounded by two concentric circles having center at z a can always be expanded into such a Laurent series (see Problem 16.92). It is possible to de ne various types of singularities of a function f z from its Laurent series. For example, when the principal part of a Laurent series has a nite number of terms and a n 6 0 while a n 1 ; a n 2 ; . . . are all zero, then z a is a pole of order n. If the principal part has in nitely many terms, z a is called an essential singularity or sometimes a pole of in nite order.
EXAMPLE. 1 1 The function e1=z 1 has an essential singularity at z 0. z 2! z2
FUNCTIONS OF A COMPLEX VARIABLE
[CHAP. 16
BRANCHES AND BRANCH POINTS Another type of singularity is a branch point. These points play a vital role in the construction of single-valued functions from ones that are multiple-valued, and they have an important place in the computation of integrals. In the study of functions of a real variable, domains were chosen so that functions were singlevalued. This guaranteed inverses and removed any ambiguities from di erentiation and integration. The applications of complex variables are best served by the approach illustrated below. It is in the realm of real variables and yet illustrates a pattern appropriate to complex variables. p Let y2 x; x > 0, then y x. In real variables two functions f1 and f2 are described by p p y x on x > 0, and y x on x > 0, respectively. Each of them is single-valued. An approach that can be extended to complex variable results by de ning the positive x-axis (not including zero) as a cut in the plane. This creates two branches f1 and f2 of a new function on a domain called the Riemann axis. The only passage joining the spaces in which the branches f1 and f2 , respectively, are de ned is through 0. This connecting point, zero, is given the special name branch point. Observe that two points x in the space of f1 and x in that of f2 can appear to be near each other in the ordinary view but are not from the Riemannian perspective. (See Fig. 16-1.)
Fig. 16-1
The above real variables construction suggests one for complex variables illustrated by w z1=2 . In polar coordinates e2i 1; therefore, the general representation of w z1=2 in that system is w 1=2 ei  2k =2 , k 0; 1. Thus, this function is double-valued. If k 0, then w1 1=2 ei=2 , 0 <  2;  > 0 If k 1, then w2 1=2 ei  2 =2 1=2 ei=2 i  1=2 ei=2 ; 2 <  4;  > 0. Thus, the two branches of w are w1 and w2 , where w1 w2 . (The double valued characteristic of w is illustrated by noticing that as z traverses a circle, C: jzj  through the values  to 2. The functional values run from 1=2 ei=2 to 1=2 ei . In other words, as z navigates the entire circle, the range variable only moves halfway around the corresponding range circle. In order for that variable to complete the circuit, z would have to make a second revolution. Thus, we would have coincident positions of z giving rise to distinct values of w. For example, z1 e =2 =i and z2 e =2 2 i are coincident points on the unit p p 2 2 1=2 1=2 1 i and z2 1 i . circle. The distinct functional values are z1 2 2 The following abstract construction replaces the multiple-valued function with a new single-valued one. Make a cut in the complex plane that includes all of the positive x-axis except the origin. Think of two planes, P1 and P2 , the rst one of in nitesimal distance above the complex plane and the other in nitesimally below it. The point 0 which connects these spaces is called a branch point. The planes
CHAP. 16]
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