s3_95-100a

S3_95-100a - e ¯ z is not analytic(b Let f z be an analytic function of z Show that the function ¯ f defined by ¯ f z = f(¯ z is also an

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ACM95a/100a October 14, 2009 Problem Set III 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.) Consider the complex function f ( z ) = u + iv = x 3 (1+ i ) - y 3 (1 - i ) x 2 + y 2 for z 6 = 0 0 for z = 0 Show that the partial derivatives of u and v with respect to x and y exist at z = 0 and that u x = v y and u y = - v x there: the Cauchy-Riemann equations are satisfied at z = 0. On the other hand, show that lim z 0 f ( z ) z does not exist, that is, f is not complex-differentiable at z = 0. Does this example contradict our theory? Explain. 2. (20 pts.) Show that the function f ( z ) = e - z - 4 for z 6 = 0 0 for z = 0 satisfies the Cauchy-Riemann equations everywhere, including at z = 0, but f ( z ) is not analytic at the origin. Does this example contradict our theory? Explain. [Hint: Consider the ray z = re iπ/ 4 .] 3. (10 pts.) Analyticity and conjugation: (a) Show that the function
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Unformatted text preview: e ¯ z is not analytic. (b) Let f ( z ) be an analytic function of z . Show that the function ¯ f defined by ¯ f ( z ) = f (¯ z ) is also an analytic function of z . 4. (20 pts.) Show that the functions defined by f ( z ) = log | z | + i arg( z ) and f ( z ) = p | z | e i arg( z ) / 2 are holomorphic (analytic) in the sector | z | > 0, | arg( z ) | < π . What are the corresponding derivatives df/dz ? 5. (30 pts.) Show that the following functions are harmonic. For each one of them find its harmonic conjugate and form the corresponding holomorphic function. (a) u ( x,y ) = x ln( r )-y arctan( y x ) ( r 6 = 0). (b) u ( x,y ) = arg( z ) ( | arg( z ) | < π , r 6 = 0). (c) u ( x,y ) = r n cos( nθ ). (d) u ( x,y ) = y/r 2 ( r 6 = 0). Due at 5pm on Wed. October 21....
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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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