New PDF release: A First Course in Complex Analysis with Applications

By Dennis G. Zill

ISBN-10: 0763714372

ISBN-13: 9780763714376

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

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Extra resources for A First Course in Complex Analysis with Applications

Example text

48. When z is a point within the open disk defined by |z| < 4, an upper bound for z 3 − 2z 2 + 6z + 2 is given by . 49. 6 we saw that if z1 is a root of a polynomial equation with real coefficients, then its conjugate z2 = z¯1 is also a root. Assume that the cubic polynomial equation az 3 + bz 2 + cz + d = 0, where a, b, c, and d are real, has exactly three roots. One of the roots must be real because . 50. 29. 29 Figure for Problem 50 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, ...

33. Suppose z is a complex number that possesses a fourth root w that is neither real nor pure imaginary. Explain why the remaining fourth roots are neither real nor pure imaginary. 34. Suppose z = r(cos θ + i sin θ) is complex number such that 1 < r < 2 and 0 < θ ≤ π/4. Suppose further that w0 is a cube root of z corresponding to k = 0. 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.

Using (9) with r = 2, θ = 7π/6, and n = 3 we get √ − 3−i 3 = 23 cos 3 7π 6 + i sin 3 7π 6 since cos(7π/2) = 0 and sin(7π/2) = –1. = 8 cos 7π 7π + i sin = −8i 2 2 20 Chapter 1 Complex Numbers and the Complex Plane Note in Example 3, if we also want the value of z −3 , then we could proceed in two ways: either find the reciprocal of z 3 = −8i or use (9) with n = −3. de Moivre’s Formula When z = cos θ + i sin θ, we have |z| = r = 1, and so (9) yields n (cos θ + i sin θ) = cos nθ + i sin nθ. (10) This last result is known as de Moivre’s formula and is useful in deriving certain trigonometric identities involving cos nθ and sin nθ.

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


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