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Complex Variables Demysti ed in Java
Complex Variables Demysti ed Creating QR Code ISO/IEC18004 In Java Using Barcode drawer for Java Control to generate, create QR Code 2d barcode image in Java applications. QR Code 2d Barcode Recognizer In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. Since the exponential function is entire, the cosine and sin functions are also entire Other derivatives follow from elementary calculus: d tan z = sec 2 z dz d sec z = sec z tan z dz d cot z = csc 2 z dz d csc z = csc z cot z dz (454) (455) Creating Bar Code In Java Using Barcode creator for Java Control to generate, create bar code image in Java applications. Bar Code Reader In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. The derivatives of the hyperbolic functions can be derived easily using exponential representations: d cosh z = sinh z dz d tanh z = sech 2 z dz d sech z = sech z tanh z dz Finally, we note the derivative of a complex exponent: d z = z 1 dz (459) d sinh z = cosh z dz (456) (457) (458) Making QR Code In Visual C#.NET Using Barcode creator for .NET framework Control to generate, create Denso QR Bar Code image in .NET framework applications. Denso QR Bar Code Generation In .NET Framework Using Barcode drawer for ASP.NET Control to generate, create QRCode image in ASP.NET applications. Note, however, that since this is a multivalued function, this holds for z > 0, 0 < arg z < 2 or some other interval Generate QRCode In .NET Using Barcode printer for .NET framework Control to generate, create QRCode image in .NET framework applications. QR Drawer In VB.NET Using Barcode generation for .NET framework Control to generate, create QR Code image in VS .NET applications. Branches
Make UCC  12 In Java Using Barcode printer for Java Control to generate, create UCC  12 image in Java applications. Create ANSI/AIM Code 39 In Java Using Barcode encoder for Java Control to generate, create Code 39 image in Java applications. A multivalued function repeats itself when z moves in a complete circle about the origin in the complex plane When 0 < 2 , the function is single valued We say that we are on one branch of the function But as we let z traverse the circle again so we enter the region where 2 < , the function repeats We say that we ve entered another branch of the function A multivalued function like this repeats itself any number of times For convenience, a barrier is set up at our choosing in the complex plane where we do not allow z to cross This barrier is called a branch cut The point from which the branch cut originates is called a branch point The branch cut extends out from the branch point to in nity For example, for a multivalued function, we can take the branch point to be the origin and the branch cut can extend out from the origin to positive in nity (Fig 416) Make EAN / UCC  13 In Java Using Barcode maker for Java Control to generate, create EAN / UCC  13 image in Java applications. Printing Bar Code In Java Using Barcode maker for Java Control to generate, create barcode image in Java applications. Elementary Functions
Paint ISSN  10 In Java Using Barcode maker for Java Control to generate, create ISSN image in Java applications. Scanning Code 3 Of 9 In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. Figure 416 Some multivalued functions repeats themselves after z has completely gone around the origin We prevent the function from being multivalued by staying on one branch This means we cannot cross the branch cut, which we have chosen in this case to be the line from the origin to positive in nity Note that a circle does not have to be used, we just have to let z go completely around the origin a circle was used here for simplicity Scan UCC.EAN  128 In Visual C#.NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in .NET applications. Painting European Article Number 13 In Visual Basic .NET Using Barcode generation for .NET Control to generate, create GTIN  13 image in Visual Studio .NET applications. Summary Decode Code 128 In Visual Basic .NET Using Barcode decoder for .NET Control to read, scan read, scan image in Visual Studio .NET applications. Barcode Reader In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. In this chapter, we described the basic properties of some elementary functions encountered in complex variables These included polynomials, the complex exponential, the trig functions, the logarithm, the hyperbolic functions, and functions with complex exponents Generating Linear Barcode In Visual Basic .NET Using Barcode generation for VS .NET Control to generate, create Linear 1D Barcode image in VS .NET applications. ECC200 Encoder In None Using Barcode maker for Software Control to generate, create ECC200 image in Software applications. Quiz
1 Prove that cos( x + iy) = cos x cos iy sin x sin iy 2 If f ( x ) = e x, then f can never be negative Is the same true of e z 3 Find a compact expression for e2+3 i 4 Find an identity for 1 + tan 2 z by using Eq (412) 5 Find an identity for tan( z + w) 6 Are the inverse trig functions multivalued This page intentionally left blank
Sequences and Series
It is common practice and often a necessity to represent a function of a real variable using an in nite series expansion It turns out that this is also true when working with complex functions As we will see, there are some new concepts involved when working with complex functions We begin by considering sequences Sequences
Consider the positive integers n = 1, 2, 3, and consider a function on the positive integers, which we denote by f (n) We call such a function a sequence The output of the function is a number: f (n) = an So a sequence is an ordered set of numbers a1 , a2 , a3 , and we refer to an as the nth term in the sequence Sequences can also be indicated using curly braces, so we can write { f (n)} or {an }

