 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
qr code generator vb.net Exponential Form of a Complex Number in VS .NET
Exponential Form of a Complex Number Recognize Code 128C In Visual Studio .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET applications. Code 128 Code Set C Creator In .NET Framework Using Barcode drawer for .NET Control to generate, create Code 128C image in .NET framework applications. In this section we introduce the exponential ( expo NEN shal ) form of a complex number. We do this because certain operations can be greatly simpli ed if the complex numbers are written in this form. It is necessary, however, in order to give a proper explanation of the ORIGIN of the exponential form, that we make use of certain procedures from more advanced mathematics. In our explanations we ll state the facts as clearly as we can, so that you can understand, in a good general way, the origin of the exponential form. It is important that we do this, because it will give you added con dence in handling complex numbers in this form, which, as you ll be pleased to nd, is not at all hard to do. Let us proceed as follows. The ratio of the circumference of a circle to the diameter is certainly one of the best known, and most used, numbers in mathematics. It is a constant ratio, universally represented by the Greek letter (pi), being an irrational number* having the approximate value 3:141 592 65: : : . Another number, equal in importance to , also exists. This number is denoted by the Greek letter (epsilon) and, like , is an irrational number, having the approximate value 2:718 281 828 459: : : . The number represented by arises in the study of the logarithmic function and, for a very speci c reason, is de ned to be equal to 1 n 2:718 28 : : : 146 lim 1 n!1 n Using your calculator, you can verify, for example, the following approximate values of (rounded o to ve decimal places): n 10 n 100 n 1000 n 10 000 1:1 10 1:01 100 1:001 1000 2:593 74 2:704 81 2:716 92 Decoding Code 128 In .NET Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET applications. Barcode Encoder In VS .NET Using Barcode generator for Visual Studio .NET Control to generate, create barcode image in .NET applications. 1:0001 10 000 2:718 15 Bar Code Decoder In VS .NET Using Barcode reader for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. Code 128B Generation In Visual C#.NET Using Barcode printer for .NET Control to generate, create Code 128A image in .NET framework applications. Thus, as n becomes in nitely great does not become in nitely great, but becomes an in nite string of nonrepeating decimals, having a limiting value of something less than 2.72. As mentioned above, the number arises in the study of the logarithmic function. In regard to logarithms, any system of logarithms has a base number, which let us denote by b. Now let y be any positive number; the logarithm of y is then de ned to be the POWER that the base number b must be raised to, to equal y. That is, the statement that 147 logb y x Code 128 Code Set C Encoder In .NET Framework Using Barcode generation for ASP.NET Control to generate, create Code 128 image in ASP.NET applications. Make Code 128 Code Set C In Visual Basic .NET Using Barcode creator for Visual Studio .NET Control to generate, create Code 128B image in VS .NET applications. * See note 11 in Appendix.
Barcode Drawer In .NET Framework Using Barcode generation for Visual Studio .NET Control to generate, create barcode image in Visual Studio .NET applications. UCC128 Printer In VS .NET Using Barcode printer for Visual Studio .NET Control to generate, create GS1 128 image in .NET applications. @Spy
Barcode Printer In .NET Using Barcode encoder for Visual Studio .NET Control to generate, create bar code image in .NET framework applications. Paint Ames Code In .NET Framework Using Barcode maker for .NET Control to generate, create USS Codabar image in .NET framework applications. means that
Reading Code 128A In Visual Studio .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. Generate GS1128 In Java Using Barcode generation for Android Control to generate, create EAN128 image in Android applications. CHAPTER 6 Algebra of Complex Numbers
Encoding Bar Code In VB.NET Using Barcode printer for Visual Studio .NET Control to generate, create barcode image in Visual Studio .NET applications. Print Bar Code In Java Using Barcode creator for Android Control to generate, create bar code image in Android applications. y bx 148
Bar Code Encoder In ObjectiveC Using Barcode printer for iPhone Control to generate, create bar code image in iPhone applications. Code39 Scanner In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. (where logb y is read as the logarithm of y to the base b ). The reason the foregoing is important is that the expression in eq. (146) appears at a critical point in the development of the logarithmic and exponential functions in the calculus. It is found that the formulas of the calculus, in regard to logarithmic and exponential work, are much simplifed if is used as the base number. Hence, in advanced mathematics, it is always understood that is the logarithmic base number; thus, for this case eq. (148) becomes y x 149 Code39 Drawer In None Using Barcode maker for Software Control to generate, create Code 39 Extended image in Software applications. Drawing 1D In Visual Basic .NET Using Barcode encoder for VS .NET Control to generate, create Linear image in VS .NET applications. which is the fundamental form of the exponential function as it appears in advanced mathematics. Next, eq. (149) can be written in the equivalent form of a power series in x;* thus x 1 x x2 x3 x4 x5 xn 2! 3! 4! 5! n! 150 in which the exclamation mark ! denotes the PRODUCT of all the positive integers from 1 to n inclusive; thus n! 1 2 3 4 n hence, 2! 1 2 2, 3! 1 2 3 6, 4! 1 2 3 4 24, and so on. The symbol ! can be read as factorial ; thus we have 2 factorial, 3 factorial, and so on. In the study of power series it is shown that the series form of eq. (150) is a valid representation of x for all positive and negative values of the variable x. Next, the sine and cosine functions can also be expressed in the power series form. Thus the following relationships are valid for all positive and negative values of x: sin x x cos x 1 x3 x5 x7 3! 5! 7! x2 x4 x6 2! 4! 6! 151 152 which, it should be noted, are valid in the above form only if angle x is measured in radians. (The series become awkward to write for x in degrees.) These are said to be alternating series, because the terms are alternately positive and negative in sign, as you can see. Now let us return to eq. (150) and boldly take the step of replacing x with jx. Assuming this is permissible (which it is), and upon taking careful note of the values of the powers of j from section 6.1, you should nd that eq. (150) becomes ! ! x2 x4 x6 x3 x5 x7 jx 1 j x 2! 4! 6! 3! 5! 7! Now compare the last equation with eqs. (151) and (152); doing this, we see that jx cos x j sin x

