s4_95-100a

s4_95-100a - f z and z t applying the chain rule of...

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ACM95a/100a October 21, 2009 Problem Set IV 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. (50 pts.) With reference to pages 129 through 131 of the Saff and Snider text, find solutions to the five boundary value problems for the Laplace equation depicted in Figures 3.11, 3.12, 3.13, 3.14 and 3.16, and evaluate them at the points (0 , 0), (0 , 0), (1 , 1), (2 , 3) and (0 , 0), respectively. 2. (20 pts.) Suppose that a function f ( z ) is analytic at a point z 0 = z ( t 0 ) on a differentiable arc z = z ( t ) ( a t b ). Show that if w ( t ) = f [ z ( t )], then w 0 ( t ) = f 0 [ z ( t )] z 0 ( t ) for all t sufficiently close to t = t 0 . Carry out your proof by expressing the real and imaginary parts of w ( t ) in terms of the real and imaginary parts of the functions
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Unformatted text preview: f ( z ) and z ( t ), applying the chain rule of calculus for functions of two variables, and using the Cauchy-Riemann equations. 3. (20 pts.) Evaluate each one of the following two integrals in two different ways: ( a ) Z 2 1 (1 /t-i ) 2 dt and ( b ) Z π/ 6 e 2 it dt. 4. (10 pts.) In view of the properties of the exponential function we have Z π e (1+ i ) x dx = Z π e x cos xdx + i Z π e x sin xdx. Evaluate the two integrals on the right by evaluating the single integral on the left and then identifying the real and imaginary parts of the values found. Verify your answers by computing the two integrals on the right-hand side by means of an alternative procedure. Due at 5pm on Wed. October 28....
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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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