Get A First Course in Complex Analysis with Applications PDF

By Dennis G. Zill

ISBN-10: 0763714372

ISBN-13: 9780763714376

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

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1+ 27. 1 2 29. √ 3i 9 10 + 12 i √ √ π π 2 cos + i 2 sin 8 8 12 26. (2 − 2i)5 √ √ 28. (− 2 + 6i)4 √ 2π 2π 30. 3 cos + i sin 9 9 6 In Problems 31 and 32, write the given complex number in polar form and in then in the form a + ib. π π π 12 π 5 31. cos + i sin 2 cos + i sin 9 9 6 6 3π 3π + i sin 8 8 π π 2 cos + i sin 16 16 3 8 cos 32. 10 33. Use de Moivre’s formula (10) with n = 2 to find trigonometric identities for cos 2θ and sin 2θ. 34. Use de Moivre’s formula (10) with n = 3 to find trigonometric identities for cos 3θ and sin 3θ.

Consider the set S of points in the complex plane defined by {i/n} , n = 1, 2, 3, . . Discuss which of the following terms apply to S: boundary, open, closed, connected, bounded. 42. Consider a finite set S of complex numbers {z1 , z2 , z3 , . . , zn }. Discuss whether S is necessarily bounded. Defend your answer with sound mathematics. 43. A set S is said to be convex if each pair of points P and Q in S can be joined by a line segment P Q such that every point on the line segment also lies in S.

A) For z = 1, verify the identity 1 + z + z2 + · · · + zn = 1 − z n+1 . 1−z (b) Use part (a) and appropriate results from this section to establish that 1 + cos θ + cos 2θ + · · · + cos nθ = sin n + 12 θ 1 + 2 sin 12 θ for 0 < θ < 2π. The foregoing result is known as Lagrange’s identity and is useful in the theory of Fourier series. 50. Suppose z1 , z2 , z3 , and z4 are four distinct complex numbers. 4 that –2 and 2 are from algebra z1 − z2 z3 − z4 = π . 2 Recall said to be square roots of the number 4 because (−2)2 = 4 and (2)2 = 4.

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


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