02 - ASSIGNMENT 2 REVISED SOLUTIONS MAT 572 A FALL 2007...

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ASSIGNMENT 2 · REVISED SOLUTIONS MAT 572 A · FALL 2007 Problem 1 ( cf. [1, Exercise III.2.9]) . Prove that the principal branch of the logarithm is continuous. Proof. Let G = C \ { z | z = < ( z ) 0 } , and suppose r n exp( n ) r exp( ) in G with r n and r positive and ϑ n and ϑ in ( - π, π ). We need to show that log r n + n log r + in C , and for this it suffices to show that r n r and ϑ n ϑ . Since | · | is continuous, the first is easy: r n = | r n exp( n ) | → | r exp( ) | = r. Now r 6 = 0, so we have exp( n ) = r n exp( n ) r n r exp( ) r = exp( ) . Using an Euler formula, it follows that cos ϑ n cos ϑ and sin ϑ n sin ϑ. The proof is finished in three cases. Suppose first that sin ϑ = 0. Then ϑ = 0, so cos ϑ n cos ϑ = 1, so eventually we have cos ϑ n > 0. It follows that eventually - π/ 2 < ϑ n < π/ 2, and then, using the continuity of sin - 1 : [ - 1 , 1] [ - π/ 2 , π/ 2], 1 we get ϑ
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02 - ASSIGNMENT 2 REVISED SOLUTIONS MAT 572 A FALL 2007...

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