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The Behavior of a Complex Function at Infinity

In: Selecta Mathematica

Author

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  • Karl Menger

    (Illinois Institute of Technology)

Abstract

Traditionally, the behavior at ∞ of a complex function f is defined as the behavior at 0 of the function obtained by substituting the —1st power into f. This definition adequately describes the class of values f(z) for large z. For instance, the range near ∞of the identity function j (whose value for any z is j(z)=z) coincides with the range near 0 of the function j -1. But that definition does not in any way describe the structural behavior of f near ∞, reflected in properties of the class of pairs (z, f (z)) for large z. In fact, the association of the value f(z) with z may, by the substitution of j -1, completely change its character. For instance, the derivative of j near ∞ is the constant function 1, whereas that j -1 of near 0 goes even faster to ∞ than does j -1

Suggested Citation

  • Karl Menger, 2003. "The Behavior of a Complex Function at Infinity," Springer Books, in: Bert Schweizer & Abe Sklar & Karl Sigmund & Peter Gruber & Edmund Hlawka & Ludwig Reich & Leopold Sc (ed.), Selecta Mathematica, pages 47-48, Springer.
    • Handle: RePEc:spr:sprchp:978-3-7091-6045-9_6
    DOI: 10.1007/978-3-7091-6045-9_6
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