Complex Variables

Gerald Dennis Mahan
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Gerald Dennis Mahan: Pennsylvania State University

Chapter 4 in Applied Mathematics, 2002, pp 73-118 from Springer

Abstract: Abstract Complex variables use the imaginary constant $$i = \sqrt { - 1.} $$ Before discussing the variables, we will review some properties of imaginary numbers. First of all, note that 4.1 $${i^2} = - 1$$ 4.2 $${i^3} = - i$$ 4.3 $${i^4} = 1$$ A fundamental relationship is 4.4 $${e^{i\theta }} = \cos (\theta ) + i\sin (\theta )$$ Using this formula, some formulas at key angles are 4.5 $${e^{i\pi /2}} = i$$ 4.6 $${e^{i\pi /2}} = {i^2} = - 1$$ 4.7 $${e^{3i\pi /2}} = {i^3} = - i$$ 4.8 $${e^{2i\pi }} = {i^4} = 1$$ 4.9 $${e^{i\pi /4}} = \sqrt i = {{1 + i} \over {\sqrt 2 }}$$ 4.10 $${e^{i\pi /6}} = {i^{1/3}} = {{\sqrt 3 + i} \over 2}$$

Keywords: Saddle Point; Real Axis; Branch Point; Complex Variable; Meromorphic Function (search for similar items in EconPapers)
Date: 2002
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DOI: 10.1007/978-1-4615-1315-5_4

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