This strange and energetic textbook bargains a transparent and intuitive method of the classical and lovely thought of complicated variables. With little or no dependence on complex options from several-variable calculus and topology, the textual content makes a speciality of the actual complex-variable rules and methods. available to scholars at their early levels of mathematical research, this complete first yr direction in advanced research deals new and engaging motivations for classical effects and introduces similar subject matters stressing motivation and method. a number of illustrations, examples, and now three hundred routines, enhance the textual content. scholars who grasp this textbook will emerge with an exceptional grounding in advanced research, and a pretty good knowing of its large applicability.

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**Example text**

Z 6 = 1. b. z 4 = −1. √ c. z 4 = −1 + 3i. 13. Show that the n-th roots of 1 (aside from 1) satisfy the “cyclotomic” equation z n−1 + z n−2 + · · · + z + 1 = 0. ] 14. Suppose we consider the n − 1 diagonals of a regular n-gon inscribed in a unit circle obtained by connecting one vertex with all the others. Show that the product of their lengths is n. ] 20 1 The Complex Numbers 15. Describe the sets whose points satisfy the following relations. Which of the sets are regions? z−1 = 1. a. |z − i| ≤ 1.

N + 2)! Cn+2 z 2 + · · · , 2! 11, if a power series is equal to zero throughout a neighborhood of the origin, it must be identically zero. For then all its derivatives at the origin–and hence all the coefﬁcients of the power series–would equal 0. By the same reasoning, if a power series were equal to zero throughout an interval containing the origin, it would be identically zero. An even stronger result is proven below. 12 Uniqueness Theorem for Power Series n Suppose ∞ n=0 Cn z is zero at all points of a nonzero sequence {z k } which converges to zero.

Show that the n-th roots of 1 (aside from 1) satisfy the “cyclotomic” equation z n−1 + z n−2 + · · · + z + 1 = 0. ] 14. Suppose we consider the n − 1 diagonals of a regular n-gon inscribed in a unit circle obtained by connecting one vertex with all the others. Show that the product of their lengths is n. ] 20 1 The Complex Numbers 15. Describe the sets whose points satisfy the following relations. Which of the sets are regions? z−1 = 1. a. |z − i| ≤ 1. b. z+1 c. |z − 2| > |z − 3|. e. d. |z| < 1 and Im z > 0.