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Unformatted text preview: (a) sin z = 2 i (b) cos z = √ 2 (c) cos z = i sin z 6. Find the domain of analyticity of the given function and compute its derivative. (a) f ( z ) = Log( z-3 + i ) (b) f ( z ) = Log( z + 4) z 2 + i 7. Find a branch of log( z 2-5 z + 3) that is analytic at z = 1 and compute its derivative there. 8. Find a branch of log( z 2 + 1) that is analytic at z = 0 and takes the value 2 πi there. Moreover, compute its derivative at z = 0. 9. Find a branch of the function (4 + z 2 ) 1 / 2 that is analytic outside the disc | z | ≤ 2. Bonus Show that the complex function f ( z ) = | z | 2 sin ± 1 | z | ² , z 6 = 0 , , z = 0 , is diﬀerentiable at z = 0 and that the ﬁrst partials u x ,u y ,v x ,v y of its real and imaginary parts exist and satisfy the Cauchy-Riemann equations at (0 , 0), but that u x and u y are not continuous at (0 , 0). 2...
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This note was uploaded on 05/16/2011 for the course MATH 332 taught by Professor Jt during the Spring '10 term at Waterloo.
- Spring '10