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Singular Values of Meromorphic Functions

In: Value Distribution of Meromorphic Functions

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  • Jianhua Zheng

    (Tsinghua University, Department of Mathematical Sciences)

Abstract

This chapter is devoted to discussing singular values of a transcendental meromorphic function. The singular value is that in any neighborhood of which the inverse of the function contains a multiple-valued branch. A value is a singular value if and only if it is an asymptotic value or a critical value. We show the construction of the parabolic simply connected Riemann surface associated with a fixed meromorphic function, and point out that every boundary point of the Riemann surface is an asymptotic value of the function. Next we consider dense properties of singularities of the inverse of a meromorphic function including relationships among singular values and between the number of direct singularities and the growth order. We then exhibit Eremenko’s construction of a meromorphic function with every value on the extended complex plane as its asymptotic value. Finally, we discuss the existence of (repelling) fixed-points of a meromorphic function of finite type, that is, the set of its singular values is bounded, and consider the case when the singular values do not distribute along a sequence of annuli.

Suggested Citation

  • Jianhua Zheng, 2010. "Singular Values of Meromorphic Functions," Springer Books, in: Value Distribution of Meromorphic Functions, chapter 0, pages 229-266, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-12909-4_6
    DOI: 10.1007/978-3-642-12909-4_6
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