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FUNCTIONS OF A COMPLEX VARIABLE in Visual Studio .NET
FUNCTIONS OF A COMPLEX VARIABLE QR Code 2d Barcode Recognizer In Visual Studio .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in VS .NET applications. Quick Response Code Generator In VS .NET Using Barcode drawer for .NET Control to generate, create Quick Response Code image in Visual Studio .NET applications. and the connecting point constitute a Riemann surface, and w1 and w2 are the branches of the function each de ned in one of the planes. (Since the space of complex variables is the complex plane, this Riemann surface may be thought of as a ight of fancy that supports a rigorous analytic construction.) To visualize this Riemann surface and perceive the singlevalued character of the new function in it, rst think of duplicates, C1 and C2 of the domain circle, C: jzj in the planes P1 and P2 , respectively. Start at on C1 , and proceed counterclockwise to the edge U2 of the cut of P1 . (This edge corresponds to 2). Paste U2 to L1 , the initial edge of the cut on P2 . Transfer to P2 through this join and continue on C2 . Now after a complete counterclockwise circuit of C2 we reach the edge L2 of the cut. Pasting L2 to U1 provides passage back to P1 and makes it possible to close the curve in the Riemann plane. See Fig. 162. QR Code JIS X 0510 Scanner In .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in Visual Studio .NET applications. Barcode Encoder In Visual Studio .NET Using Barcode drawer for Visual Studio .NET Control to generate, create bar code image in VS .NET applications. Fig. 162 Decode Bar Code In Visual Studio .NET Using Barcode recognizer for VS .NET Control to read, scan read, scan image in .NET framework applications. Printing Quick Response Code In C#.NET Using Barcode printer for VS .NET Control to generate, create Quick Response Code image in .NET framework applications. Note that the function is not continuous on the positive xaxis. Also the cut is somewhat arbitrary. Other rays and even curves extending from the origin to in nity can be employed. In many integration applications the cut i proves valuable. On the other hand, the branch point (0 in this example) is special. If another point, z0 6 0 were chosen as the center of a small circle with radius less than jz0 j, then the origin would lie outside it. As a point z traversed its circumference, its argument would return to the original value as would the value of w. However, for any circle that has the branch point as an interior point, a similar traversal of the circumference will change the value of the argument by 2, and the values of w1 and w2 will be interchanged. (See Problem 16.37.) QR Code JIS X 0510 Encoder In Visual Studio .NET Using Barcode encoder for ASP.NET Control to generate, create QR image in ASP.NET applications. Making QR Code JIS X 0510 In Visual Basic .NET Using Barcode encoder for .NET framework Control to generate, create Quick Response Code image in .NET framework applications. RESIDUES The coe cients in (9) can be obtained in the customary manner by writing the coe cients for the Taylor series corresponding to z a n f z . In further developments, the coe cient a 1 , called the residue of f z at the pole z a, is of considerable importance. It can be found from the formula a 1 lim 1 d n 1 f z a n f z g n 1 ! dzn 1 10 2D Barcode Generator In VS .NET Using Barcode printer for VS .NET Control to generate, create Matrix Barcode image in Visual Studio .NET applications. Painting GS1128 In VS .NET Using Barcode drawer for .NET Control to generate, create UCC128 image in Visual Studio .NET applications. where n is the order of the pole. For simple poles the calculation of the residue is of particular simplicity since it reduces to a 1 lim z a f z Code 128B Generation In .NET Framework Using Barcode generator for .NET Control to generate, create Code 128A image in Visual Studio .NET applications. ITF14 Generation In .NET Using Barcode generator for Visual Studio .NET Control to generate, create UCC  14 image in Visual Studio .NET applications. 11 Print Matrix 2D Barcode In C# Using Barcode creator for .NET Control to generate, create 2D Barcode image in VS .NET applications. USS Code 39 Encoder In VS .NET Using Barcode creator for ASP.NET Control to generate, create Code 39 Full ASCII image in ASP.NET applications. RESIDUE THEOREM If f z is analytic in a region r except for a pole of order n at z a and if C is any simple closed curve in r containing z a, then f z has the form (9). Integrating (9), using the fact that UPCA Supplement 5 Recognizer In C# Using Barcode scanner for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Barcode Creator In ObjectiveC Using Barcode encoder for iPhone Control to generate, create bar code image in iPhone applications. FUNCTIONS OF A COMPLEX VARIABLE
European Article Number 13 Generation In ObjectiveC Using Barcode maker for iPad Control to generate, create EAN / UCC  13 image in iPad applications. Paint Barcode In .NET Framework Using Barcode creator for Reporting Service Control to generate, create barcode image in Reporting Service applications. [CHAP. 16
Reading Barcode In C# Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET framework applications. Universal Product Code Version A Creation In None Using Barcode generation for Online Control to generate, create UCC  12 image in Online applications. dz z a n C
0 2i
if n 6 1 if n 1
12 (see Problem 16.13), it follows that f z dz 2ia 1
13 i.e., the integral of f z around a closed path enclosing a single pole of f z is 2i times the residue at the pole. More generally, we have the following important theorem. Theorem. If f z is analytic within and on the boundary C of a region r except at a nite number of poles a; b; c; . . . within r, having residues a 1 ; b 1 ; c 1 ; . . . ; respectively, then f z dz 2i a 1 b 1 c 1 14 i.e., the integral of f z is 2i times the sum of the residues of f z at the poles enclosed by C. Cauchy s theorem and integral formulas are special cases of this result, which we call the residue theorem. EVALUATION OF DEFINITE INTEGRALS The evaluation of various de nite integrals can often be achieved by using the residue theorem together with a suitable function f z and a suitable path or contour C, the choice of which may reuqire great ingenuity. The following types are most common in practice. 1 F x dx; F x is an even function. 1. 0 F z dz along a contour C consisting of the line along the xaxis from R to Consider R and the semicircle above the xaxis having this line as diameter. Problems 16.29 and 16.30. 2 G sin ; cos d, G is a rational function of sin and cos . Then let R ! 1. z z 1 z z 1 ; cos and dz iei d or d dz=iz. The Then sin 2i 2 given integral is equivalent to F z dz, where C is the unit circle with center at the origin. See Let z ei . Problems 16.31 and 16.32. & ' 1 cos mx F x dx; F x is a rational function. sin mx 1 F z eimz dz where C is the same contour as that in Type 1. Here we consider Problem 16.34. 4. Miscellaneous integrals involving particular contours. See Problems 16.35 and 16.38. In particular, Problem 16.38 illustrates a choice of path for an integration about a branch point. CHAP. 16]

