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# T2sol - Complex Variables MAT334 T ERM TEST 2 Solution 1(5...

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Complex Variables MAT334 T ERM TEST 2 Solution 1. (5 points) State the Liouville’s theorem. Ans 1: Let f be an entire function. If there exists a real number M > 0 such that | f ( z ) | < M for all z C then f must be constant on C . 2. (10 points) Res e π 2 iz ( z 2 + 1)( z - 1) , 1 = e π 2 i (1 + 1) = i 2 Here we used that the given function has a pole of order 1 at 1, since the function e π 2 iz / ( z 2 + 1) is analytic at 1 and its value at 1 is not 0. Res cos z e z - 1 , 0 = cos 0 e 0 = 1 Here we used the fact that if for some functions F ( z ) and G ( z ) , F ( z 0 ) = 0 , G ( z 0 ) = 0 , G ( z 0 ) = 0 and both F and G are analytic at z 0 then Res ( F G , z 0 ) = F ( z 0 ) G ( z 0 ) . We applied this fact to F ( z ) = cos z and G ( z ) = e z - 1 at z 0 = 0 . 3. (10 points) Let f ( z ) = z sin z ( z - π ) 3 . Find the first three terms of the Laurent series of the function at z = π . Determine the residue of f at π . Ans: Let h ( z ) = z sin z . To determine the Laurent series for f we first find the (first few terms of the) Taylor series for h at z = π . To do that we compute: h ( z ) = z sin z h ( π ) = 0 h ( z ) = sin z + z cos z h ( π ) = - π h ( z ) = 2 cos z - z sin z h ( π ) = - 2 h ( z ) = -

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T2sol - Complex Variables MAT334 T ERM TEST 2 Solution 1(5...

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