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Unformatted text preview: A. State the definition of a residue. Ans: If the complex function f has an isolated singularity (+1 point) at the point z , then f has a Laurent series representation (+1 point) ( 29 ( 29 ( 29 ( 29 2 1 1 2 ...... ...... k k k a a f z a z z a a z z z z z z ∞-- =-∞ =- = + + + +- +-- ∑ which is valid in some deleted neighborhood z z R <- < (+2 points) . The coefficient 1 a- of 1 z z- (+2 points) is called the residue of f at z . (At most 5 points) Line-circle preserving property Ans: A linear fractional transformation (+1 point) maps a circle in the z-plane to either a line or a circle in the w-plane (+2 points) . The image is a line if and only if the original circle passes through a pole of the linear fractional transformation (+2 points) . B. True or false (If it is false, explain briefly why it isn’t true). (15%) 1. The only possible singularities of a rational function are poles....
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- Spring '07