Finalreview

# Finalreview - MAT 461 Some Review For Final Posted December...

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MAT 461 - Some Review For Final Posted December 4 These problems are meant to help supplement your studying. They will cover many, but not all of the topics that may appear on the final. 1. Sketch the sets of points determined by the following conditions. Are these regions (open, closed or neither), (bounded or unbounded), (connected or not connected)? • | z + 2 - i | = 2 • | z - 3 i | ≤ 3 • | z - 1 | = | z - i | 2. Assuming that | z | = 1, show that vextendsingle vextendsingle vextendsingle vextendsingle 2 z + 1 z 2 + 4 iz vextendsingle vextendsingle vextendsingle vextendsingle 1 3. Let z = 2 - 2 i . Put z in exponential form. Find z 4 . Find the roots z 1 2 . Give the roots in principal argument form. Which is the principal root? Find Log z (Log z has branch cut at - π ). Simplify 1 ¯ z into x + iy form. 4. Let f ( z ) = 3 z 2 - 1 z ( z - i ) . Compute the following limits: lim z 0 f ( z ) lim z 1 f ( z ) lim z →∞ f ( z ) 5. Use the Cauchy-Riemann equations to check whether the following are analytic. ( f ( z ) = u ( x, y ) + iv ( x, y )).

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