s8_95-100a

# s8_95-100a - unit circle under the boundary conditions = 1...

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ACM95a/100a November 25, 2009 Problem Set VIII 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. (20 pts.) Find a M¨obius transformation mapping the lower half-plane to the disk | w + 1 | < 1. [Hint: Do it in steps.] 2. (20 pts.) Map the region between the two cirles | z | = 2 and | z - 1 | = 1 conformally onto the upper half plane. [Hint: Use a M¨ obius transformation to map the point 2 to . Show that the image region is a strip . Then use the exponential map, after modifying (translating, rotating) the strip, if necessary.] 3.- (30 pts.) Find the electrostatic potential in the part of the upper half plane exterior to the
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Unformatted text preview: unit circle under the boundary conditions = 1 for { ( x,y ) : x 2 + y 2 = 1 and y } , and = 0 for { ( x,y ) : y = 0 and | x | > 1 } . [Hint: consider the map w = z + 1 /z and recall your solution of problem 1, Figure 3.14, in Problem Set IV.] 4.- (30 pts.) Find the electrostatic potential in the slit upper half-plane, or, more precisely, solve the problem = 0 for y > 0, except the part of the y-axis axis between (0 , 0) and (0 , 1) = 0 on y = 0 = 1 on the part of the y-axis axis between (0 , 0) and (0 , 1). [Hint: Note that the map z z 2 maps both the real and the imaginary axes into the real line.] Due at 5pm on Wed. December 2nd....
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## This note was uploaded on 01/14/2010 for the course ACM 1 taught by Professor Prof during the Fall '09 term at Caltech.

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