Exam (elaborations) TEST BANK FOR Advanced Engineering Mathematics [Volume 2] By Herbert Kreyszig and Erwin Kreyszig (Student Solutions Manual and Study Guide)

Complex numbers appeared in the textbook before in different topics. Solving linear homogeneous ODEs led to characteristic equations, (3), p. 54 in Sec. 2.2, with complex numbers in Example 5, p. 57, and Case III of the table on p. 58. Solving algebraic eigenvalue problems in Chap. 8 led to characteristic equations of matrices whose roots, the eigenvalues, could also be complex as shown in Example 4, p. 328. Whereas, in these type of problems, complex numbers appear almost naturally as complex roots of polynomials (the simplest being x2 C1 D 0), it is much less immediate to consider complex analysis—the systematic study of complex numbers, complex functions, and “complex” calculus. Indeed, complex analysis will be the direction of study in Part D. The area has important engineering applications in electrostatics, heat flow, and fluid flow. Further motivation for the study of complex analysis is given on p. 607 of the textbook. We start with the basics in Chap. 13 by reviewing complex numbers z D x C yi in Sec. 13.1 and introducing complex integration in Sec.13.3. Those functions that are differentiable in the complex, on some domain, are called analytic and will form the basis of complex analysis. Not all functions are analytic. This leads to the most important topic of this chapter, the Cauchy–Riemann equations (1), p. 625 in Sec. 13.4, which allow us to test whether a function is analytic. They are very short but you have to remember them! The rest of the chapter (Secs. 13.5–13.7) is devoted to elementary complex functions (exponential, trigonometric, hyperbolic, and logarithmic functions). Your knowledge and understanding of real calculus will be useful. Concepts that you learned in real calculus carry over to complex calculus; however, be aware that there are distinct differences between real calculus and complex analysis that we clearly mark. For example, whereas the real equation ex D 1 has only one solution, its complex counterpart ez D 1 has infinitely many solutions. 258 Complex Analysis Part D Sec. 13.1 Complex Numbers and Their Geometric Representation Much of the material may be familiar to you, but we start from scratch to assure everyone starts at the same level. This section begins with the four basic algebraic operations of complex numbers (addition, subtraction, multiplication, and division). Of these, the one that perhaps differs most from real numbers is division (or forming a quotient). Thus make sure that you remember how to calculate the quotient of two complex numbers as given in equation (7), Example 2, p. 610, and Prob. 3. In (7) we take the number z2 from the denominator and form its complex conjugate Nz2 and a new quotient Nz2=Nz2. We multiply the given quotient by this new quotient Nz2=Nz2 (which is equal to 1 and thus allowed): z D z1 z2 D z1 z2
1 D z1 z2
Nz2 Nz2 ; which we multiply out, recalling that i 2 D