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# pmidterm - z i z 4-z Problem 10 Find lim n →∞ sinh in n...

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MAT334, Midterm Practice Problems Problem 1. Compute ( 2 / 2 + i 2 / 2) 239 . Problem 2. Compute exp(ln 2 + πi ). Problem 3. Find all the solutions of the equation z 5 = 32. Problem 4. Find lim z 0 e z - e - z z Problem 5. Show that the function u ( x, y ) = sin x ( e y + e - y ) is harmonic, and find its harmonic conjugate. Problem 6. Show that the function f ( x + iy ) = x 4 + y 4 + 1 - 2 x 2 + 2 y 2 - 6 x 2 y 2 + i (4 x 3 y - 4 xy 3 - 4 xy ) is entire, and find its derivative. Corrected! Problem 7. For a function f ( z ) = z + i z - i find the level curve | f ( z ) | = 1. Problem 8. Write the polynomial z 4 - z + 1 at the Taylor form centered at i . Problem 9. Find all the poles of the rational function

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Unformatted text preview: z + i z 4-z . Problem 10. Find lim n →∞ sinh( in ) n . 1 Answers Problem 1. √ 2 / 2-i √ 2 / 2. Problem 2.-2. Problem 3. 1, e 2 πi/ 5 , e 4 πi/ 5 , e 6 πi/ 5 , e 8 πi/ 5 . Problem 4. 2 - use L’Hospital rule. Problem 5. cos x ( e y-e-y ). Problem 6. f ± = 4 x 3-4 x-12 xy 2 + i (12 x 2 y-4 y 3-4 y ). Problem 7. Imz = 0. Problem 8. 2-i-(4 i + 1)( z-i )-6( z-i ) 2 + 4 i ( z-i ) 3 + ( z-i ) 4 . Problem 9.-i , 0, 1, e 2 πi/ 3 , e 4 πi/ 3 . Problem 10....
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pmidterm - z i z 4-z Problem 10 Find lim n →∞ sinh in n...

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