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.
Read more about this topic: Pole (complex Analysis)
Famous quotes containing the words pole and/or infinity:
“No exile at the South Pole or on the summit of Mont Blanc separates us more effectively from others than the practice of a hidden vice.”
—Marcel Proust (1871–1922)
“As we begin to comprehend that the earth itself is a kind of manned spaceship hurtling through the infinity of space—it will seem increasingly absurd that we have not better organized the life of the human family.”
—Hubert H. Humphrey (1911–1978)