hw2 - Complex Analysis Spring 2001 Homework II Due Friday...

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Unformatted text preview: Complex Analysis Spring 2001 Homework II Due Friday April 20 1. With defined to be the least positive zero of cos t, we established in class Wednesday 2 that t eit was onto the first quadrant of the unit circle in C. Use proven properties of the complex exponential to prove that cos( - t) = - cos t, cos( + t) = - cos t, sin( - t) = sin t, and sin( + t) = - sin t and use these to finish the proof that eit is onto. 2. Conway, chapter 3, section 2, problem 6. 3. Conway, chapter 3, section 2, problem 19. 4. Conway, chapter 3, section 2, problem 21. 5. Find all real numbers a, b, c, d such that the real polynomial u = ax3 + bx2 y + cxy 2 + dy 3 is harmonic. Find a harmonic conjugate in the unit disk two ways: by the integration process starting at the origin of the plane; and secondly by finding directly an analytic function whose real part is the given function. 6. Let w = f (z) = z i be defined using the principal branch of the logarithm. Describe the image in the w-plane of circles centered at 0 in the z-plane and rays from 0 in the z-plane (omitting points z on the negative real axis, of course). ...
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