This strange and full of life textbook deals a transparent and intuitive method of the classical and lovely thought of complicated variables. With little or no dependence on complex ideas from several-variable calculus and topology, the textual content makes a speciality of the genuine complex-variable principles and methods. available to scholars at their early levels of mathematical examine, this complete first yr path in complicated research deals new and engaging motivations for classical effects and introduces similar themes stressing motivation and procedure. a variety of illustrations, examples, and now three hundred workouts, improve the textual content. scholars who grasp this textbook will emerge with an outstanding grounding in advanced research, and a superior realizing of its large applicability.

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So suppose N x + y − 2i x y ≡ 2 αk (x + i y)k . 2 k=0 Setting y = 0, we obtain N x2 ≡ αk x k k=0 or α0 + α1 x + (α2 − 1)x 2 + · · · + α N x N ≡ 0. Setting x = 0 gives α0 = 0; dividing out by x and again setting x = 0 shows α1 = 0, etc. We conclude that α1 = α3 = α4 = · · · = α N = 0 α2 = 1, and so our assumption that N x 2 + y 2 − 2i x y ≡ αk (x + i y)k k=0 has led us to x 2 + y 2 − 2i x y ≡ (x + i y)2 = x 2 − y 2 + 2i x y, which is simply false! A bit of experimentation, using the method described above (setting y = 0 and “comparing coefﬁcients”) will show how rare the analytic polynomials are.

Sin2 z + cos2 z = 1, c. (sin z) = cos z. * Show that a. sin( π2 + iy) = 12 (e y + e−y ) = cosh y b. | sin z| ≥ 1 at all points on the square with vertices ±(N + 12 )π ± (N + 12 )π i, for any positive integer N. c. | sin z| → ∞, as Imz = y → ±∞. Exercises 43 17. Find (cos z) . 18. Find sin−1 (2)– that is, ﬁnd the solutions of sin z = 2. * Find all solutions of the equation: z ee = 1. 20. Show that sin(x + iy) = sin x cosh y + i cos x sinh y. 21. Show that the power series f (z) = 1 + z + z2 + ...

Power Series Expansion about z = α All of the previous results on power series are easily adapted to power series of the form Cn (z − α)n . By the simple substitution w = z − α, we see, for example, that series of the above form converge in a disc of radius R about z = α and are differentiable throughout |z − α| < R where R = 1/lim|Cn |1/n . ) Exercises 1. 3 by showing that for an analytic polynomial P, Py = i Px . * a. Suppose f (z) is real-valued and differentiable for all real z. Show that f (z) is also real-valued for real z.