Selecta Mathematica pp 47-48 | Cite as

# The Behavior of a Complex Function at Infinity

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## 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}