93下複變期末考 電&a

93&auml&ced - w = cos z(8(5 Construct a linear fractional transformation that maps the triple z 1 =-1 z 2 = 0 z 3 = 1 to the triple w 1 = i

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Complex Analysis - Final Examination 10:20AM to 12:20 PM, June 21, 2005 (1) (10 %) Expand f ( z )= e z in a Taylor series centered at z =3 i . Give the radius of convergence. (2) (10 %) Expand f ( z )= 1 z ( z - 3) in a Laurent series valid in 1 < | z - 4 | < 4. (3) (10 %) Determine the order of the poles for f ( z )=tan z . (4) (15 %) Use Cauchy’s residue theorem to evaluate ± C z 3 e - 1 /z 2 dz along (a) | z | = 5, (b) | z + i | = 2, (c) | z - 3 | =1. (5) (10 %) Evaluate the Cauchy principal value of ² -∞ x sin x ( x 2 +1) dx . (6) (10 %) Find a complex mapping from R = { z = x + iy | 0 y π } to R ± = { w = u + iv | 0 Arg( w ) 3 π/ 2 } . (7) (10 %) Use the identity cos z = sin( π/ 2 - z ) to ±nd the image of R = { z = x + iy | 0 x π } under the complex mapping
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Unformatted text preview: w = cos z . (8) (5 %) Construct a linear fractional transformation that maps the triple ( z 1 =-1 , z 2 = 0 , z 3 = 1) to the triple ( w 1 = i, w 2 = ∞ , w 3 = 0). (9) (10 %) Use the Schwarz-Christoffel formula to construct a conformal mapping from the upper half plane R = { z = x + iy | y ≥ } to R ± = { w = u + iv | u ≥ 0 and 0 ≤ v ≤ π } . Require that f (-1) = iπ and f (1) = 0. (10) (10 %) Prove that P . V . ² ∞ x a-1 1-x dx = π cot aπ with 0 < a < 1. 1...
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This note was uploaded on 02/21/2012 for the course EE 101 taught by Professor 張捷力 during the Spring '07 term at National Taiwan University.

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