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

By D. J. H. Garling

The 3 volumes of A direction in Mathematical research supply a whole and precise account of all these parts of actual and intricate research that an undergraduate arithmetic pupil can count on to come across of their first or 3 years of analysis. Containing enormous quantities of workouts, examples and functions, those books becomes a useful source for either scholars and lecturers. quantity I makes a speciality of the research of real-valued services of a true variable. quantity II is going directly to think about metric and topological areas. This 3rd quantity covers advanced 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

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We label certain points of ∂R as follows: N = 2i, S = −2i, E = 1, W = −1, N E = 1 + 2i, SE = 1 − 2i, N W = −1 + 2i, SW = −1 − 2i. We call [N W, N E] the top of ∂R, and [SW, SE] the bottom of ∂R. The set ∂R is the track of a Jordan curve, and of course the Jordan curve theorem holds for it; let U be the inside of ∂R and V the outside. If γ : [a, b] → C is a path with γ(a) ∈ U and γ(b) ∈ V then γ −1 (U ) and γ −1 (V ) are disjoint non-empty open subsets of [a, b]. 3 The Jordan curve theorem 663 γ −1 (∂R) is a non-empty closed subset of [a, b].

Let C = {s ∈ [0, 1] : [hs ] = [β]}. Use a compactness argument to show that C is closed. Use the intermediate value theorem and a compactness argument to show that C is open. Deduce that β is not null-homotopic, and that Π1 (U, (0, 0)) is not commutative. 3 The Jordan curve theorem The results of the previous section now enable us to prove one of the famous results of mathematics. To conform with tradition, we shall call a simple closed path in C a Jordan curve, although, in the terminology used in Volume II, it need not be a curve.

2 Suppose that γ : [a, b] → C is a rectifiable path and that f is a continuous complex-valued function on [γ]. Then there exists a unique complex number Iγ (f ) with the property that if > 0 then there exists δ > 0 such that if D = (a = t0 < · · · < tn = b) is a dissection of [a, b] with mesh size less than δ then |Iγ (f ) − SD (f ; γ)| ≤ 2 . Proof Let Dn be the dissection of [a, b] into 2n intervals of equal length. Then Dm is a refinement of Dn , for m > n, and the mesh size of Dn tends to 0 as n → ∞.

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