Titlebook: Complex Analysis with Applications to Number Theory; Tarlok Nath Shorey Textbook 2020 Springer Nature Singapore Pte Ltd. 2020 Cauchy Theor

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The Prime Number Theorem with an Error Term,Rewriting PNT as ., it is clear from Theorem . that . is a better approximation than . to .. We have given a proof of PNT in the previous chapter by a method different from that of J. Hadamard and de la Vallée Poussin.

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The Baker Theorem,A complex number . is called .  if there exists a non-zero polynomial . such that . and . if . such that .(.) is monic and .. In fact . satisfies the unique polynomial .(.) of minimal degree with relatively prime integer coefficients such that the leading coefficient of . is positive.

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Complex Analysis with Applications to Number Theory978-981-15-9097-9Series ISSN 2363-6149 Series E-ISSN 2363-6157

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Introduction and Simply Connected Regions,ce. We introduce the notion of connectedness in Sect. 1.2, and we show in Theorem . that an open set in a metric space is a disjoint union of open connected sets which we call .. Further, we prove Theorem . in Sect. 1.3 that the extended complex plane is a metric space homeomorphic to the Riemann sp

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Harmonic Functions,unction are harmonic. We prove the existence of a harmonic conjugate of a harmonic function in a simply connected region in Sect. 4.4 where we also prove its converse. We introduce continuous functions with Mean Value Property in a region . in Sect. 4.5 and prove in Sect. 4.5 Maximum principle for h

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