exsheet4c - Sivakumar M407 Example Sheet 4c 1 Let log z...

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Unformatted text preview: Sivakumar M407 Example Sheet 4c 1. Let log( z ) denote a branch of the logarithm defined on the region D := C \{ x (1+ i ) : x ≥ } . Assume that log( i ) = 5 πi/ 2. (i) Evaluate log(- 1), log(- 2 i ), log(1- i ), and log(3). (ii) Find the image of the lower-half plane H ↓ := { z ∈ C : = ( z ) < } under the mapping z 7→ log( z ). 2. (i) Find all possible values of (- 1) 1 /π . (ii) Find the principal value of [( e/ 2)(- 1- √ 3 i )] 3 πi . (iii) Find all possible values of (- 1 + √ 3 i ) 3 / 2 . 3. Let c = a + ib be a fixed complex number, where c is not an integer. Find all possible values of i c . What restriction must be placed on the constant c so that the values of | i c | are all the same? 4. Show, by means of an example, that Log( z 1 z 2 ) 6 = Log( z 1 ) + Log( z 2 ) in general. 5. (i) Suppose that α is a fixed real number. Show that | z α | = e α ln( | z | ) for every nonzero complex number z ....
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exsheet4c - Sivakumar M407 Example Sheet 4c 1 Let log z...

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