homework20111118

homework20111118 - f z ±nd the order of the zero at z = 0...

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Math 417, Fall 2011 Exercise Set #8 Turn in: November 18 1. For each of the following functions, ±nd all of the isolated singularities. Determine whether they are removable singularities, poles or essential singularities. If the function has a pole, determine the order N of the pole. ( a ) 1 z 3 +1 ( b ) sin( z 2 ) z 2 ( c ) cos( z ) 1 z 4 ( d )(tan( z )) 4 ( e ) z 2 e z 1 2. Find the residues of f ( z ) at all the isolated singular points in the complex plane for the following functions: ( a ) z +5 z ( z 2 +1) ( b ) 2 z +1 ( z 1) 2 ( z +1) ( c ) z 2 e 1 /z 3. Let C be the positively oriented simple closed contour | z | =3 . Ca l cu la t eth e following integrals. ( a ) ° C ze z z 2 1 dz ( b ) ° C cosh( πz ) z ( z 2 +1) dz ( c ) ° C z 3 e 1
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Unformatted text preview: f ( z ), ±nd the order of the zero at z = 0. ( a ) cos( z 2 ) − 1 ( b ) sin( z 3 ) ( c ) ( e z − 1) 3 ( d ) z 7 − 12 z 5 + z 3 5. a) Suppose that f ( z ) is an entire function, with a pole at z = ∞ of order m ∞ ≥ 1. Show that f ( z ) is a polynomial of degree m ∞ . b) Suppose that f ( z ) is analytic at all points of C −{ z } , and f ( z ) has a pole of order m at z = z and a pole of order m ∞ at z = ∞ . Show that f ( z ) = P ( z ) Q ( z ) where P ( z ) and Q ( z ) are polynomials of ±nite degree. 1...
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