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Functions of a Complex Variable in .NET
Functions of a Complex Variable Scan QR Code 2d Barcode In VS .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in VS .NET applications. Print Denso QR Bar Code In .NET Using Barcode printer for .NET framework Control to generate, create Denso QR Bar Code image in Visual Studio .NET applications. Ultimately it was realized that to accept numbers that provided solutions to equations such as x2 1 0 was no less meaningful than had been the extension of the real number system to admit a solution for x 1 0, or roots for x2 2 0. The complex number system was in place around 1700, and by the early nineteenth century, mathematicians were comfortable with it. Physical theories took on a completeness not possible without this foundation of complex numbers and the analysis emanating from it. The theorems of the di erential and integral calculus of complex functions introduce mathematical surprises as well as analytic re nement. This chapter is a summary of the basic ideas. Recognize QR Code In Visual Studio .NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET applications. Print Bar Code In .NET Framework Using Barcode generator for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. FUNCTIONS If to each of a set of complex numbers which a variable z may assume there corresponds one or more values of a variable w, then w is called a function of the complex variable z, written w f z . The fundamental operations with complex numbers have already been considered in 1. A function is singlevalued if for each value of z there corresponds only one value of w; otherwise it is multiplevalued or manyvalued. In general, we can write w f z u x; y iv x; y , where u and v are real functions of x and y. Recognizing Bar Code In .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in VS .NET applications. Creating QR Code ISO/IEC18004 In C# Using Barcode encoder for Visual Studio .NET Control to generate, create QR Code image in Visual Studio .NET applications. EXAMPLE. w z2 x iy 2 x2 y2 2ixy u iv so that u x; y x2 y2 ; v x; y 2xy. called the real and imaginary parts of w z2 respectively. These are QR Code JIS X 0510 Maker In VS .NET Using Barcode creator for ASP.NET Control to generate, create QR image in ASP.NET applications. QR Code Drawer In Visual Basic .NET Using Barcode maker for .NET Control to generate, create Quick Response Code image in .NET framework applications. In complex variables, multiplevalued functions often are replaced by a specially constructed singlevalued function with branches. This idea is discussed in a later paragraph. UPC Symbol Generator In .NET Framework Using Barcode generation for .NET Control to generate, create GS1  12 image in .NET applications. 2D Barcode Maker In VS .NET Using Barcode printer for .NET framework Control to generate, create Matrix 2D Barcode image in VS .NET applications. EXAMPLE. Since e2ki 1, the general polar form of z is z ei 2k . This form and the fact that the logarithm and exponential functions are inverse leads to the following de nition of ln z ln z ln 2k i k 0; 1; 2; . . . ; n . . . Making Bar Code In .NET Framework Using Barcode creation for Visual Studio .NET Control to generate, create barcode image in VS .NET applications. British Royal Mail 4State Customer Code Generator In .NET Framework Using Barcode generator for .NET framework Control to generate, create RoyalMail4SCC image in .NET applications. Each value of k determines a singlevalued function from this collection of multiplevalued functions. These are the branches from which (in the realm of complex variables) a singlevalued function can be constructed. EAN13 Printer In Visual Studio .NET Using Barcode encoder for Reporting Service Control to generate, create EAN13 image in Reporting Service applications. Code 3 Of 9 Printer In Java Using Barcode generation for Java Control to generate, create Code 39 Extended image in Java applications. Copyright 2002, 1963 by The McGrawHill Companies, Inc. Click Here for Terms of Use.
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ANSI/AIM Code 39 Printer In None Using Barcode maker for Online Control to generate, create Code39 image in Online applications. EAN13 Decoder In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. LIMITS AND CONTINUITY De nitions of limits and continuity for functions of a complex variable are analogous to those for a real variable. Thus, f z is said to have the limit l as z approaches z0 if, given any > 0, there exists a > 0 such that j f z lj < whenever 0 < jz z0 j < . Similarly, f z is said to be continuous at z0 if, given any > 0, there exists a > 0 such that j f z f z0 j < whenever jz z0 j < . Alternatively, f z is continuous at z0 if lim f z f z0 . z!z0 Note: While these de nitions have the same appearance as in the real variable setting, remember that jz z0 j < means q j x x0 j i y y0 j x x0 2 y y0 2 < : Thus there are two degrees of freedom as x; y ! x0 ; y0 : DERIVATIVES If f z is singlevalued in some region of the z plane the derivative of f z , denoted by f 0 z , is de ned as z!0 f z z f z z
provided the limit exists independent of the manner in which z ! 0. If the limit (1) exists for z z0 , then f z is called analytic at z0 . If the limit exists for all z in a region r, then f z is called analytic in r. In order to be analytic, f z must be singlevalued and continuous. The converse, however, is not necessarily true. We de ne elementary functions of a complex variable by a natural extension of the corresponding functions of a real variable. Where series expansions for real functions f x exists, we can use as de nition the series with x replaced by z. The convergence of such complex series has already been considered in 11. EXAMPLE 1. We de ne ex 1 z z2 z3 z3 z5 z7 z2 z4 z6 ; sin z z ; cos z 1 . 2! 3! 3! 5! 7! 2! 4! 6! ex cos y i sin y , as well as numerous other relations.

