232sheet3 - B U Department of Mathematics Math 232...

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B U Department of Mathematics Math 232 Introduction To Complex Analysis Spring 2008 Exercise Sheet 3 1 1. A complex number z = x + iy may also be visualised as a 2 × 2 matrix ± x y - y x ² Verify that addition and multiplication of complex numbers defined via matrix operations are con- sistent with the usual addition and multiplication rules. What is the matrix representation corre- sponding to z - 1 ? 2. A complex number of the form α + , where α,β Z \ { 0 } is called a Gaussian integer . A Gaussian integer a is said to be composite if it can be factored into the form a = bc , where b and c are both Gaussian integers (excluding ± 1, ± i ); otherwise it is said to be prime. Show that, as Gaussian integers, 2 is composite but 3 is prime. 3. Consider the circline given by the equation ³ ³ ³ ³ z - α z - β ³ ³ ³ ³ = λ , ( λ > 0). We know, if λ = 1 we have a line, say L , and the points α and β are reflections of each other in L . Now, let λ 6 = 1. Prove that there are exactly two points z 1 , z
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232sheet3 - B U Department of Mathematics Math 232...

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