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Important Mathematical Results: Schwarz Reflection Principle, Sugawara–Kanazawa Theorem, and Herglotz Theorem

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Part of the book series:SpringerBriefs in Physics ((SpringerBriefs in Physics))

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Abstract

TheSchwarz reflection principle states that given a functionf(z) of a complex variablez such thatf(z) is real in a finite segment\(\Gamma \) of the real axis.

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Notes

  1. 1.

    The notation of Ref. [15] for a infinite definite limit refers to the fact that the function should tend to this infinite limit steadily without oscillating.

  2. 2.

    In the manipulations for the demonstration of this theorem we take that the vanishing limits for\(x\rightarrow \infty \) tend to zero at least as\(x^{-\gamma }\) with\(\gamma >0\).

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Authors and Affiliations

  1. Departamento de Física, Universidad de Murcia, Murcia, Spain

    José Antonio Oller

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  1. José Antonio Oller

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Correspondence toJosé Antonio Oller.

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© 2019 The Author(s), under exclusive licence to Springer Nature Switzerland AG

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Oller, J.A. (2019). Important Mathematical Results: Schwarz Reflection Principle, Sugawara–Kanazawa Theorem, and Herglotz Theorem. In: A Brief Introduction to Dispersion Relations. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-13582-9_4

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