A First Course in Complex Analysis with Applications by Dennis G. Zill

By Dennis G. Zill

Written for junior-level undergraduate scholars which are majoring in math, physics, machine technological know-how, and electric engineering.

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Extra resources for A First Course in Complex Analysis with Applications

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10. arg(−2 + 10i) = π − tan−1 (5) + 2nπ for n an integer. 11. If z is a root of a polynomial equation an z n + an−1 z n−1 + · · · + a1 z + a0 = 0, then z¯ is also a root. 12. For any nonzero complex number z, there are an infinite number of values for arg(z). 13. If |z − 2| < 2, then | Arg(z) | < π/2. 14. The set S of complex numbers z = x + iy whose real and imaginary parts are related by y = sin x is a bounded set. 15. The set S of complex numbers z satisfying |z| < 1 or | z − 3i | < 1 is a domain.

X 28. Suppose n denotes a nonnegative integer. Determine the values of n such that z n = 1 possesses only real solutions. Defend your answer with sound mathematics. 29. (a) Proceed as in Example 2 to find the approximate values of the two square roots w0 and w1 of 1 + i. 14 Figure for Problem 27 (b) Show that the exact values of the roots in part (a) are √ √ √ 1+ 2 2−1 1+ 2 +i , w1 = − −i w0 = 2 2 2 √ 2−1 . 2 30. Discuss: What geometric significance does the result in Problem 24 have? 31. Discuss: A real number can have a complex nth root.

Moreover, for z = 0 we write z/0 = ∞ and for z = ∞, z/∞ = 0. Expressions such as ∞ − ∞, ∞/∞, ∞0 , and 1∞ cannot be given a meaningful definition and are called indeterminate. 24 The method of correspondence in (b) is a stereographic projection. 5 Answers to selected odd-numbered problems begin on page ANS-4. In Problems 1–12, sketch the graph of the given equation in the complex plane. 1. |z − 4 + 3i| = 5 2. |z + 2 + 2i| = 2 3. |z + 3i| = 2 4. |2z − 1| = 4 5. Re(z) = 5 6. Im(z) = −2 34 Chapter 1 Complex Numbers and the Complex Plane 8.

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