Department of Mathematics

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    Mittag-Leffler Theorem and Series Representation of Meromorphic Functions
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Muhammad Waseem Akram; CIIT/FA19-RMT-002/LHR; Dr. Muhammad Faisal Nadeem; LHR TP 7404
    The goal of this thesis is to present a complete proof of Mittag-Leffler theorem, and to present some applications. We first give a simpler proof, valid for open disks and the whole complex plane. This proof uses in a crucial way the approximation of holomorphic function by means of polynomials on such domains, by means of their Taylor series. The proof of Mittag-Leffler theorem for an arbitrary domain G ⊂ ℂ is much more involved and requires the development of a more general theory of approximation of holomorphic function on complex domains. We present this theory and prove Runge’s theorem, a fundamental result in complex analysis providing conditions to have the required approximations. This allows us to complete the proof of the general version of Mittag-Leffler theorem. We complete this thesis by presenting some applications of the above results.
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    Analytical Continuation and Natural Domains of Holomorphic Functions of One Variable
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Muhammad Fiaz; CIIT/FA19-RMT-033/LHR; Dr. Muhammad Aqeel Ahmad Khan; LHR TP 7423
    In this thesis, we will talk about analytic continuations of holomorphic functions. We will compare real and complex settings. We know that holomorphic functions can be represent as a Taylor series in their domains. We will find the region outside of the domains with the help of analytic continuation where these functions are holomorphic. We will introduce the bump function in order to prove the analytic continuations of real smooth functions is not unique. On the other hand, we will see that analytic continuations of holomorphic functions is unique along a chosen curve. In general, it is not true that analytic continuation is unique along two different curves. It will happen when domain is simply connected and if function can be continued along every curves with common initial and end points in its domain. We will prove a core result of analytic continuation known as Monodromy Theorem. We will also see singular points of holomorphic functions and some results on extensions of holomorphic functions on bounded domains.