Pole At Infinity
A complex function can be defined as having a pole at the point at infinity. In this case U has to be a neighborhood of infinity, such as the exterior of any closed ball. To use the previous definition, a meaning for g being holomorphic at ∞ is needed. Alternately, a definition can be given starting from the definition at a finite point by suitably mapping the point at infinity to a finite point. The map does that. Then, by definition, a function f holomorphic in a neighborhood of infinity has a pole at infinity if the function (which will be holomorphic in a neighborhood of ), has a pole at, the order of which will be regarded as the order of the pole of f at infinity.
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