# Complex Analysis in One Variable by Raghavan Narasimhan

By Raghavan Narasimhan

This e-book offers complicated research in a single variable within the context of recent arithmetic, with transparent connections to a number of complicated variables, de Rham thought, actual research, and different branches of arithmetic. therefore, protecting areas are used explicitly in facing Cauchy's theorem, actual variable equipment are illustrated within the Loman-Menchoff theorem and within the corona theorem, and the algebraic constitution of the hoop of holomorphic features is studied.

Using the original place of complicated research, a box drawing on many disciplines, the booklet additionally illustrates strong mathematical principles and instruments, and calls for minimum heritage fabric. Cohomological equipment are brought, either in reference to the lifestyles of primitives and within the learn of meromorphic functionas on a compact Riemann floor. The evidence of Picard's theorem given the following illustrates the powerful regulations on holomorphic mappings imposed through curvature conditions.

New to this moment variation, a set of over a hundred pages worthy of routines, difficulties, and examples supplies scholars a chance to consolidate their command of advanced research and its family members to different branches of arithmetic, together with complicated calculus, topology, and genuine applications.

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1, Proposition 6. Theorem 5 (Morera's Theorem). Let n be an open set in C and let f be a continuous junction on n. Suppose that for any closed rectangle R c n, we have [ JaR Then f is holomorphic on fdz = O. n. Proof. It is sufficient to prove the theorem when center of the disc. ) n is a disc. Let a = ex + if3 be the -+ -+ For z = n, z = x + iy(x, y E lR), let Z' = X + if3 and let yz be the curve az' . Z'Z. We define a function F on n by F(z) = 1 f(w)dw. Yz Let h f= 0 be a small real number. Then we have F(z + ih) - F(z) ---'-----'--~ h where L is the line segment z .

A holomorphic function f E H(n - E) is said to be meromorphic on n if, for any a E E, there is a disc D with center a, Den, and two functions g, h E H(D) such that h ¢' 0 on D and h . f ID - E = g ID - E. ) Lemma 3. Let D = D(a, r), D* = D - {a} and let f 00 L n=-oo Cn (z E H(D*). Let fez) - a)n be its Laurent expansion at a. Then f is meromorphic on D = if and only if there is an integer N such that Cn = Ofor n < -N. Proof· Suppose that f is meromorphic on D, and let U be a disc centered at a on which there are functions g, h E H(U), h i= 0, such that hflD* n U = glD* n U.

Conversely, suppose that f(z) = 00 L cn(z - a)n; then (z - a)N f(z) = n=-N 00 L Cn-N(Z - a)n E 1t(D) (by Abel's lemma). n=O Theorem 3. Let Q be open in C and let E C Q be a discrete set. Let f E 1t(Q - E). Then f is meromorphic on Q if, and only if, for every a E E, there exists a neighborhood U C Q with U n E = {a}, such that the following holds. Either flU - {a} is bounded (so that f extends to be holomorphic at a by Theorem 2) or If(z)1 -+- 00 as z -+- a, z #- a. Proof. Sufficiency of the condition.