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Download e-book for kindle: A Course in Mathematical Analysis, vol. 3: Complex analysis, by D. J. H. Garling

By D. J. H. Garling

ISBN-10: 1107032040

ISBN-13: 9781107032040

The 3 volumes of A path in Mathematical research offer an entire and distinctive account of all these components of genuine and complicated research that an undergraduate arithmetic pupil can count on to come across of their first or 3 years of analysis. Containing thousands of routines, examples and functions, those books becomes a useful source for either scholars and academics. quantity I specializes in the research of real-valued services of a true variable. quantity II is going directly to ponder metric and topological areas. This 3rd quantity covers complicated research and the speculation of degree and integration.

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Extra resources for A Course in Mathematical Analysis, vol. 3: Complex analysis, measure and integration

Example text

6 Suppose that f : Mr (w) → Mr (w) is continuous. Then f has a fixed point: there exists z ∈ Mr (w) with f (z) = z. Proof Without loss of generality, we can suppose that w = 0. Let g(z) = z − f (z) for z ∈ Mr (0). We must show that the equation g(z) = 0 has 658 The topology of the complex plane a solution in Mr (0). Suppose not. Let γ(t) = κr (0)(t) = reit . Let h(t) = e−it g(γ(t)), for 0 ≤ t ≤ 2π. Then |h(t) − r| = |e−it g(γ(t)) − e−it γ(t)| = |e−it (g(γ(t)) − γ(t))| = |f (γ(t))| ≤ r. Also g(z) = 0 for z ∈ Mr (0), and so h(t) = 0 for t ∈ [0, 2π].

If z ∈ C and z = w, let ρ(z) = w + r(z − w) ; |z − w| ρ(z) is the unique point in Tr (w) ∩ Rw,z , where Rw,z = {w + λ(z − w) : λ ≥ 0} is the ray from w that contains z. The mapping ρ : C \ {w} → Tr (w) is a retract of C\{w} onto Tr (w); it is the natural retract of C\{w} onto Tr (w). The restriction of ρ to the punctured neighbourhood Mr◦ (w) = Mr (w) \ {w} is also a retract, of Mr◦ (w) onto Tr (w). We cannot do better. 8 Tr (w). There does not exist a retract f of Mr (w) onto Proof If w + z ∈ Tr (w), let t(w + z) = w − z.

J=1 Thus there exists K ≥ 1 such that | kj=1 (1 + fj (x))| ≤ K for all k ∈ N and all x ∈ X. Suppose that 0 < < 1/2. There exists j0 such that ∞ j=j0 Mj < /2K. 1, l j=k+1 (1 + fj (x)) − 1 ≤ , K and so l k (1 + fj (x)) − j=1 (1 + fj (x)) j=1 k = l (1 + fj (x)) · 1 − j=1 (1 + fj (x)) < . j=k+1 Thus the products converge uniformly on X. 1 to the product of k terms, and then letting k tend to infinity. 6 The maximum modulus principle 647 whose restriction to U is analytic. If z0 ∈ U then |f (z0 )| < sup{|f (z)| : z ∈ ∂U }.

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A Course in Mathematical Analysis, vol. 3: Complex analysis, measure and integration by D. J. H. Garling


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