A first course in complex analysis by Beck M., Marchesi G., Pixton G.

By Beck M., Marchesi G., Pixton G.

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Show that a polynomial of odd degree with real coefficients must have a real zero. ) 10. Suppose f is entire and there exist constants a, b such that |f (z)| ≤ a|z| + b for all z ∈ C. Prove that f is a linear polynomial (that is, of degree ≤ 1). 11. In this problem F (z) = eiz z 2 +1 and R > 1. 2: (a) Show that σ F (z) dz = πe if σ is the counterclockwise semicircle formed by the segment S of the real axis from −R to R, followed by the circular arc T of radius R in the upper half plane from R to −R.

We do this by defining a new set, C define algebraic rules for dealing with infinity based on the usual laws of limits. For example, if lim f (z) = ∞ and lim g(z) = a is finite then the usual “limit of sum = sum of limits” rule gives z→z0 z→z0 lim (f (z) + g(z)) = ∞. This leads to the addition rule ∞ + a = ∞. We summarize these rules: z→z0 CHAPTER 3. 3. Suppose a ∈ C. (a) ∞ + a = a + ∞ = ∞ (b) ∞ · a = a · ∞ = ∞ · ∞ = ∞ if a = 0. a a = 0 and = ∞ if a = 0. ∞ 0 If a calculation involving infinity is not covered by the rules above then we must investigate the limit more carefully.

D) sin z = cosh 4. (e) cos z = 0. (f) sinh z = 0. (g) exp(iz) = exp(iz). (h) z 1/2 = 1 + i. 28. Find the image of the annulus 1 < |z| < e under the principal value of the logarithm. 29. Show that |az | = aRe z if a is a positive real constant. 30. Fix c ∈ C \ {0}. Find the derivative of f (z) = z c . 31. Prove that exp(b log a) is single-valued if and only if b is an integer. ) What can you say if b is rational? 32. Describe the image under exp of the line with equation y = x. To do this you should find an equation (at least parametrically) for the image (you can start with the parametric form x = t, y = t), plot it reasonably carefully, and explain what happens in the limits as t → ∞ and t → −∞.

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