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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Famous quotes containing the words pole and/or infinity:
“O, withered is the garland of the war,
The soldier’s pole is fallen!”
—William Shakespeare (1564–1616)
“We must not suppose that, because a man is a rational animal, he will, therefore, always act rationally; or, because he has such or such a predominant passion, that he will act invariably and consequentially in pursuit of it. No, we are complicated machines; and though we have one main spring that gives motion to the whole, we have an infinity of little wheels, which, in their turns, retard, precipitate, and sometime stop that motion.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)