An Introduction to Complex Analysis by Ravi P. Agarwal, Kanishka Perera, Sandra Pinelas

By Ravi P. Agarwal, Kanishka Perera, Sandra Pinelas

This textbook introduces the topic of complicated research to complicated undergraduate and graduate scholars in a transparent and concise manner.

Key gains of this textbook:

-Effectively organizes the topic into simply potential sections within the type of 50 class-tested lectures

- makes use of distinct examples to force the presentation

-Includes a number of workout units that motivate pursuing extensions of the fabric, every one with an “Answers or tricks” part

-covers an array of complex subject matters which enable for flexibility in constructing the topic past the fundamentals

-Provides a concise heritage of advanced numbers

An creation to advanced research may be helpful to scholars in arithmetic, engineering and different technologies. necessities contain a path in calculus.

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Extra resources for An Introduction to Complex Analysis

Example text

15. What is wrong in the following? √ √ √ 1 = 1 = (−1)(−1) = −1 −1 = i i = − 1. 16. Show that π π (1 − i)49 cos 40 + i sin 40 √ (8i − 8 3)6 10 = − √ 2. 17. Let z = reiθ and w = Reiφ , where 0 < r < R. Show that Re w+z w−z = R2 − r 2 . 18. Solve the following equations: √ √ (a). z 2 = 2i, (b). z 2 = 1 − 3i, (c). z 4 = −16, (d). z 4 = −8 − 8 3i. 19. For the root of unity z = e2πi/m , m > 1, show that 1 + z + z 2 + · · · + z m−1 = 0. 20. Let a and b be two real constants and n be a positive integer.

Clearly, every convex set is necessarily connected. Furthermore, it follows that the intersection of two or more convex sets is also convex. 1. Shade the following regions and determine whether they are open and connected: (a). {z ∈ C : −π/3 ≤ arg z < π/2}, (b). {z ∈ C : |z − 1| < |z + 1|}, √ (c). {z ∈ C : |z − 1| + |z − i| < 2 2}, √ √ (d). {z ∈ C : 1/2 < |z − 1| < 2} {z ∈ C : 1/2 < |z + 1| < 2}. 2. Let S be the open set consisting of all points z such that |z| < 1 or |z − 2| < 1. Show that S is not connected.

Therefore, the words positive and negative are never applied to complex numbers. 1. Express each of the following complex numbers in the form x + iy : √ √ (a). ( 2 − i) − i(1 − 2i), (b). (2 − 3i)(−2 + i), (c). (1 − i)(2 − i)(3 − i), 4 + 3i 1+i i 1 + 2i 2 − i (d). , (e). + , (f). + , 3 − 4i i 1−i 3 − 4i 5i √ −10 (g). (1 + 3 i) , (h). (−1 + i)7 , (i). (1 − i)4 . 2. Describe the following loci or regions: (a). |z − z0 | = |z − z 0 |, where Im z0 = 0, (b). |z − z0 | = |z + z 0 |, where Re z0 = 0, (c).

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