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By Dennis G. Zill

Written for junior-level undergraduate scholars which are majoring in math, physics, computing device 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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Carefully sketch w0 , w02 , and w03 in the complex plane. Computer Lab Assignments In Problems 35–40, use a CAS§ to first find z n = w for the given complex number and the indicated value of n. Then, using the output and the same value of n, determine whether w1/n = (z n )1/n = z. If not, explain why not. 35. 5 − i; n = 10 36. 3i; n = 5 37. 1 + 3i; n = 8 38. 2 + 2i; n = 12 √ 40. −1 + 3 i; n = 11 39. 5sections we examined some rudiments of the algebra In the and geometry of complex numbers. But we have barely scratched the surface of the subject known as complex analysis; the main thrust of our study lies ahead.

A) the numbers on unit circle |z| = 1 (b) the numbers within the open disk |z| < 1 (c) the numbers that are exterior to unit circle, that is, |z| > 1 50. 24(b) in terms of the coordinates of the point (a, b, 0) in the complex plane. Use these formulas to verify your answer to Problem 48. 6 we are going to examine a few simple applications of complex numbers. It will be assumed in the discussion that the reader has some familiarity with methods for solving elementary ordinary differential equations.

B) Is your answer to part (a) unique? Elaborate in detail. 24. If z1 is a root of a polynomial equation with nonreal coefficients, then its conjugate z2 = z¯1 is not a root. Prove this result in the case of a quadratic equation az 2 + bz + c = 0, where at least one of the coefficients a = 0, b, or c is not real. Use your work from Problem 20 and indicate at what point out we can make this conclusion. 6 Applications 43 In Problems 25 and 26, factor the given quadratic polynomial if the indicated complex number is one root.

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