EE_401_Hwk_3_091907

EE_401_Hwk_3_091907 - 5. Find the Laurent Series expansion...

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EE 401 HOMEWORK #3 Due: Wednesday, 9/2 6 /0 7 JENKINS 1. Find the Taylor series expansion for f ( z ) = z 1 2 about the point z= 1. What is the radius of convergence of the series? 2. Give the regions of convergence and divergence for: (a) z k k 2 + 1 k = 1 (b) 2 z k = 1 k 3. Given that sinh z, cosh z, e z , sin z , and cos z are all entire, show that: (a) cosh 2 z sinh 2 z = 1 (b) sin π 2 z ( ) = cos z HINT: Start from the corresponding identities for z= real, and use uniqueness of analytic functions. 4. Show that if f ( z ) has a pole of order m at z= α , then the residue at the pole is given by: a -1 = 1 ( m 1)! d m 1 dz m 1 z ( ) m f z ( ) z =
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Unformatted text preview: 5. Find the Laurent Series expansion for f z ( ) = 1 z z 1 ( ) 2 about the point z= 1. Give the region of convergence. HINTS: Try expanding 1 z about z= 1 first. You need not compute any integrals to solve this problem. 6. For the expressions in each of parts b and c of Poularikas and Seely, problem 9 (p. 983): (i) Find all poles. (ii) Find the residue of the pole nearest the origin. (iii) Find all essential singularities. Justify your answers. NOTE: Dont forget to include the point at z = ....
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