Analytic Functions pp 214-233 | Cite as

# Meromorphic Functions of Finite Order

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## Abstract

In the preceding discussion we have frequently observed that the degree of transcendency of a meromorphic function is reflected in the growth of its characteristic *T*(*r*). Considered as a class, meromorphic functions whose characteristic is of a definite order of magnitude have a more complicated structure the higher the order of growth. Thus, for instance, we have found that among other functions meromorphic for |*z|* < 1, meromorphic functions which have a bounded characteristic in the unit disk are distinguished by certain simple properties. They have, for example, definite boundary values almost everywhere on the bounding circle |*z*| = 1, which is not generally the case otherwise. The boundedness of the characteristic of a function meromorphic in the entire plane *z* ≠ ∞ implies that the function is constant. For rational functions *T*(*r*) = *O*(log *r*), while for a transcendental function the ratio *T*(*r*):log *r* is unbounded as *r* → ∞^{1}.

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## References

- 1.This will be proved in § 2 of this chapter.Google Scholar
- 1.A. Pringsheim [1], E. Lindelöf [1]. We prefer the designation “mean type” (type moyen) suggested by Lindelöf to the term “normal type” of Pringsheim.Google Scholar
- 2.G. Valiron [1].Google Scholar
- 1.
- 1.We can assume that
*w*(0) ≠ 0, ∞. This can always be achieved by dividing the function*w*by a power*z*^{α}of*z*.Google Scholar