complex - 2.003 Fall 2003 Complex Exponentials Complex...

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2.003 Fall 2003 Complex Exponentials Complex Numbers Complex numbers have both real and imaginary components. A complex number r may be expressed in Cartesian or Polar forms: r = a + jb (cartesian) = | r | e φ (polar) The following relationships convert from cartesian to polar forms: 2 Magnitude | r | = a 2 + b ± tan 1 b a > 0 a Angle φ = tan 1 b a < 0 a ± π Complex numbers can be plotted on the complex plane in either Cartesian or Polar forms Fig.1. Figure 1: Complex plane plots: Cartesian and Polar forms Euler’s Identity Euler’s Identity states that e = cos φ + j sin φ 1
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2.003 Fall 2003 Complex Exponentials This can be shown by taking the series expansion of sin , cos , and e . sin φ = φ φ 3 3! + φ 5 5! φ 7 7! + ... cos φ = 1 φ 2 2! + φ 4 4! φ 6 6! + ... e = 1 + φ 2 2! j φ 3 3! + φ 4 4! + j φ 5 5! + ... Combining cos φ + j sin φ = 1 + ( φ ) 2 2! j
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complex - 2.003 Fall 2003 Complex Exponentials Complex...

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