L14_ComplexAnalysis

L14_ComplexAnalysis - PHYS126 L14 page 1 of 5 Lecture 14...

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Unformatted text preview: PHYS126, L14: page 1 of 5 Lecture 14: Complex Variable and Analysis (Mathematic Preparation for Quantum Mechanics) Shengwang Du, 25 March 2011 1. Representation of complex numbers ( C z ∈ ) Complex number: ib a z + = with R b a ∈ & (both a and b are real numbers) where the imaginary number i is defined by 1 2 − = i . We also label a , the real part as a z = } Re{ , and b, the imaginary part as b z = } Im{ } Im{ } Re{ z i z z + = Note: in Electrical Engineering, the symbol j is used instead of i 2. Equality 2 1 1 ib a z + = 2 2 2 ib a z + = 2 1 z z = if and only if 2 1 a a = and 2 1 b b = 3. Operations Addition: ) ( ) ( 2 1 2 1 2 1 b b i a a z z + + + = + Subtraction: ) ( ) ( 2 1 2 1 2 1 b b i a a z z − + − = − Multiplication: 2 1 2 2 1 2 1 2 1 2 2 1 1 2 1 ) ( ] ][ [ b b i a b b a i a a ib a ib a z z + + + = + + = 2 1 2 1 2 1 2 1 ) ( b b a b b a i a a − + + = ) ( ) ( 2 1 2 1 2 1 2 1 a b b a i b b a a + + − = Division: 2 2 2 2 2 1 1 2 2 1 2 1 2 2 2 2 2 2 1 1 2 2 1 1 2 1 ) ( ) ( ) )( ( ) )( ( b a b a b a i b b a a ib a ib a ib a ib a ib a ib a z z + − + + = − + −...
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L14_ComplexAnalysis - PHYS126 L14 page 1 of 5 Lecture 14...

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