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FUNCTIONS OF A COMPLEX VARIABLE in .NET
FUNCTIONS OF A COMPLEX VARIABLE Decoding QR Code 2d Barcode In .NET Framework Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET applications. QRCode Generator In Visual Studio .NET Using Barcode generation for .NET framework Control to generate, create QR Code image in Visual Studio .NET applications. 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. Scanning QR Code JIS X 0510 In VS .NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in .NET applications. Drawing Barcode In .NET Framework Using Barcode printer for .NET framework Control to generate, create barcode image in .NET applications. 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 Recognize Barcode In Visual Studio .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. Encode Denso QR Bar Code In Visual C#.NET Using Barcode generation for .NET Control to generate, create QR Code ISO/IEC18004 image in .NET framework applications. 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 . Denso QR Bar Code Creation In Visual Studio .NET Using Barcode creation for ASP.NET Control to generate, create QR Code 2d barcode image in ASP.NET applications. QRCode Printer In Visual Basic .NET Using Barcode printer for Visual Studio .NET Control to generate, create QR Code image in VS .NET applications. EXAMPLE. If f z 1 , then z 3 is an isolated singularity of f z . z 3 2
Code 128B Generator In VS .NET Using Barcode maker for VS .NET Control to generate, create Code 128C image in Visual Studio .NET applications. Barcode Printer In .NET Using Barcode encoder for VS .NET Control to generate, create barcode image in .NET applications. 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. Linear Maker In Visual Studio .NET Using Barcode maker for Visual Studio .NET Control to generate, create Linear Barcode image in .NET framework applications. Drawing Identcode In VS .NET Using Barcode printer for .NET Control to generate, create Identcode image in .NET applications. POLES
Read GS1  12 In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. Create EAN13 In None Using Barcode generation for Software Control to generate, create EAN 13 image in Software applications. 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 Code 3/9 Creation In ObjectiveC Using Barcode maker for iPhone Control to generate, create Code 39 image in iPhone applications. GTIN  12 Scanner In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. 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 EAN / UCC  14 Printer In None Using Barcode generation for Software Control to generate, create EAN128 image in Software applications. Barcode Reader In C#.NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. k 1 UCC  12 Encoder In ObjectiveC Using Barcode creation for iPhone Control to generate, create USS128 image in iPhone applications. Create Barcode In Java Using Barcode maker for BIRT Control to generate, create barcode image in BIRT reports applications. 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 singlevalued functions from ones that are multiplevalued, 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 singlevalued. An approach that can be extended to complex variable results by de ning the positive xaxis (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. 161.) Fig. 161 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 doublevalued. 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 multiplevalued function with a new singlevalued one. Make a cut in the complex plane that includes all of the positive xaxis 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]

