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

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Unformatted text preview: 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 > 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 , = cos 0 e = 1 Here we used the fact that if for some functions F ( z ) and G ( z ) , F ( z ) 6 = 0 , G ( z ) = 0 , G ( z ) 6 = 0 and both F and G are analytic at z then Res ( F G ,z ) = F ( z ) G ( z ) . We applied this fact to F ( z ) = cos z and G ( z ) = e z- 1 at z = 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 ....
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T2sol - Complex Variables MAT334 T ERM TEST 2 Solution 1....

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