{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

93&auml;&cedil;‹&egrave;&curren;‡&egrave;&reg;Š&aelig;œŸ&aelig;œ&laquo;&egrave;€ƒ &eacute;›&raquo;&a

# 93ä¸‹è¤‡è®ŠæœŸæœ«è€ƒ é›»&a

This preview shows page 1. Sign up to view the full content.

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 -∞ 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 find the image of R = { z = x + iy | 0 x π
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}