By Raghavan Narasimhan

This e-book provides advanced research in a single variable within the context of recent arithmetic, with transparent connections to a number of advanced variables, de Rham concept, actual research, and different branches of arithmetic. hence, overlaying areas are used explicitly in facing Cauchy's theorem, genuine variable equipment are illustrated within the Loman-Menchoff theorem and within the corona theorem, and the algebraic constitution of the hoop of holomorphic capabilities is studied.

Using the original place of advanced research, a box drawing on many disciplines, the publication additionally illustrates robust mathematical rules and instruments, and calls for minimum heritage fabric. Cohomological tools are brought, either in reference to the lifestyles of primitives and within the examine of meromorphic functionas on a compact Riemann floor. The evidence of Picard's theorem given right here illustrates the powerful regulations on holomorphic mappings imposed by way of curvature conditions.

New to this moment version, a suite of over a hundred pages worthy of workouts, difficulties, and examples supplies scholars a chance to consolidate their command of complicated research and its relatives to different branches of arithmetic, together with complicated calculus, topology, and actual applications.

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**Additional info for Complex Analysis in One Variable**

**Example text**

If now Ken is compact, then K C 00 U D v , so that K v=l The sequence {fqk} then converges uniformly on This proves Montel's theorem. N C N U Dv for a finite N. v=l U D v , hence on K. v=l As an instance of the way in which Montel's theorem may be applied, we prove the following classical result (which is, in fact, equivalent to Montel's theorem). Theorem 3 (Vitali). Let n be a connected open set in C and let A C n be a subset which has at least one point of accumulation in n. Let {fn}n~l be a sequence of functions holomorphic on n which is uniformly bounded on any compact subset of n.

Clearly FID* = f. Another proof, not using the Laurent expansion, runs as follows. 40 Chapter I. Elementary Theory of Holomorphic Functions If f E H(D*) and zf(z) -+ 0 as z -+ 0, z i= 0, define a function g on D by g(z) = Z2 f (z), z i= 0, g(O) = O. Then g is (:-differentiable at 0 with g' (0) = 0; in fact I -(g(S) - g(O») s = l;f(S) -+ 0 as Since clearly g is (:-differentiable on D*, we have g since g(O) = g'(O) = O. Hence fez) 00 =L n=2 S -+ 0, E s i= O. H(D). Consequently ~g(n)(0)zn-2 for small z n.

Is meromorphic. Necessity of the condition. Let a E E, V = D(a, p) a disc around a with V n E = {a} on which there exist h, g E 1t(V), h ¥= Osuch thathf = g on V -{a}. We may assume that g ¥= O. From the Taylor expansions of g and h, we can write g(z) = (z -a)kcp(z), h(z) = (z - a)l1/l(z), where cp, 1/1 E 1t(V) and cp(a) #- 0, 1/I(a) #- O. If U C V is a disc D(a, r) with small enough r, then cp(z) #- 0, 1/I(z) #- 0 on U. We then have f(z) = (z - a)k-lcp(z)/1/I(z) for z E U - {a}. It then follows that if k :::: f is If(z)1 -+- 00 as z --+ a, z #- a.