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Unformatted text preview: m + n power. All other terms are higher in power than this one. (e) If f ( z ) has a pole of order m at z = 0, then f ( z 2 ) has a pole of order 2 m at z = 0. This is a true statement that follows directly from noting the Laurent Series representation of f and making the substitution z z 2 . (f) If f has an essential singularity at z and g has a pole at z , then f + g has an essential singularity at z . This is a true statement and is obvious by summing the associated Laurent Series representations of both functions. (g) If f and g have an essential singularity at z , then f ( z ) /g ( z ) has a removable singularity at z . 1 This statement is false and a counterexample is f ( z ) = sin(1 /z ), g ( z ) = cos(1 /z ) which gives f/g = tan(1 /z ) which has an essential singularity at z . 2. Verify Picards Theorem for the function cos(1 /z ) at z = 0. 2...
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- Three '09