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s2_95-100a

# s2_95-100a - z =-i log z i(1-z 2 1 2 Explain how to...

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ACM95a/100a October 7, 2009 Problem Set II Please deposit your completed homework set to by the due-time in the Firestone 303 door-slot. Please write your Grading Section Number at the top of the first page of your homework set. 1. (10 pts.) For a given real number φ , 0 φ < 2 π , find the image of the sector 0 arg( z ) < φ under the transformation w = z 4 . How large should φ be so that the w plane is covered exactly once? 2. (10 pts.) Let α 6 = 0, β 6 = 0 be two complex numbers. Show that α = for some real number t (i.e. the vectors defined by α and β are parallel) if and only if = ( α β ) = 0. 3. (25 pts.) Show that
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Unformatted text preview: ( z ) =-i log ( z + i (1-z 2 ) 1 / 2 ) . Explain how to introduce cuts to give a branch with cos-1 (0) = π/ 2. 4. (20 pts.) Show that tan-1 ( z ) = i 2 log ± i + z i-z ² and tanh-1 ( z ) = 1 2 log ± 1 + z 1-z ² . 5. (10 pts.) Determine all values of i i and log ((1 + i ) iπ ) and plot them in the complex plane. 6. (25 pts.) Determine the branch points of the function p ( z-1)( z-6)( z + 2) . Construct cuts and deﬁne a branch so that z = 4 does not lie on a cut, and such that w = 6 i when z = 4. Due at 5pm on Wed. October 14....
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