332Ass32011

332Ass32011 - (a sin z = 2 i(b cos z = √ 2(c cos z = i...

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AMATH/PMATH 332: Assignment 3 Due: Monday, 14 February, 2011 Suggested problems: Sect. 2.5: 1, 6, 8 (a), (b). Sect. 3.2: 1, 2, 5, 7, 9, 17 (a), (b), 19, 23. Sect. 3.3: 1, 5, 6, 9, 11, 13. Sect. 3.5: 1, 3, 7, 8, 10, 11, 12, 15 (a), (b). Problems to be handed in: 1. Show that if v ( x,y ) is a harmonic conjugate of u ( x,y ) in a domain D , then every harmonic conjugate of u ( x,y ) in D must be of the form v ( x,y ) + a , where a is a real constant. 2. Let f ( z ) = u ( x,y ) + iv ( x,y ) be a complex function. (a) Show, by giving an example, that even if f ( z ) is analytic in a domain D , the function g ( z ) := v ( x,y ) + iu ( x,y ) may not be analytic in D . (b) Show nonetheless that if f ( z ) is analytic in a domain D , then the function h ( z ) := v ( x,y ) - iu ( x,y ) is also analytic in D . (c) Use (b) to show that if v is a harmonic conjugate for u , then - u is a harmonic conjugate for v . 3. (a) Explain why the function f ( z ) = sin( z 2 ) + e - z + iz is entire and find its derivative. (b) Explain why the function Re ± cos z e z ² is harmonic in the whole plane. 4. Prove the following. (a) lim z 0 sin z z = 1 (b) lim z 0 cos z - 1 z = 0 5. Find all the solutions of the given equation.
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Unformatted text preview: (a) sin z = 2 i (b) cos z = √ 2 (c) cos z = i sin z 6. Find the domain of analyticity of the given function and compute its derivative. (a) f ( z ) = Log( z-3 + i ) (b) f ( z ) = Log( z + 4) z 2 + i 7. Find a branch of log( z 2-5 z + 3) that is analytic at z = 1 and compute its derivative there. 8. Find a branch of log( z 2 + 1) that is analytic at z = 0 and takes the value 2 πi there. Moreover, compute its derivative at z = 0. 9. Find a branch of the function (4 + z 2 ) 1 / 2 that is analytic outside the disc | z | ≤ 2. Bonus Show that the complex function f ( z ) = | z | 2 sin ± 1 | z | ² , z 6 = 0 , , z = 0 , is differentiable at z = 0 and that the first partials u x ,u y ,v x ,v y of its real and imaginary parts exist and satisfy the Cauchy-Riemann equations at (0 , 0), but that u x and u y are not continuous at (0 , 0). 2...
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332Ass32011 - (a sin z = 2 i(b cos z = √ 2(c cos z = i...

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