ps11 - Riemann equation u x = v y , then check if the...

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MATH 406: Problem Set 11 due Friday, April 15, 2005 1. (F 2.6.18) Compute the following integral, using the change of variables x = e t to move the lower limit (and the singularity) to -∞ : Z 0 (ln x ) 2 x 2 + 1 dx 2. Compute the following integral: Z 0 x 1 / 3 x 3 + 1 dx 3. Fisher 3.3.5(d) 4. Tell if each function is N -analytic, specifying N , then if the function is 2-analytic, check that the real and imaginary parts are biharmonic: a) F = 7 iz + ¯ z + ¯ z 2 z 2 b) F = e z + | z | 2 c) F = ( z - 12)( z +2 i )(¯ z - 1) 5. Decide if the following functions are harmonic or not by checking the first Cauchy-
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Unformatted text preview: Riemann equation u x = v y , then check if the functions is biharmonic by checking if the dierence u x-v y is harmonic (as we showed in class): a) zz 2 + 16 z 2 b) z cos z c) ze z 6. Based on the previous question, is a biharmonic function also harmonic, or not? Ex-plain. 7. Find all points at which the following mappings would not be conformal: a) w = Ez 2 + k b) w = z cos 3 z c) w = z + 1 z d) w = z 2 + 3 z 2 + i 8. Fisher 3.4.8...
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