Lect08-Complex Variables

# Lect08-Complex Variables - M.D Bryant ME 340 notes Complex...

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M.D. Bryant ME 340 notes 5/29/2009 Complex Variables z = x + j y x = Re( z ), real part of z y = Im( z ), imaginary part of z j = -1 Similar to a vector: z = 1 x + j y 1 and j like “unit vectors” Plot in complex plane x = Re z y = Im z r = | z | θ = arg z -y z = x + j y z = x - j y -

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M.D. Bryant ME 340 notes 5/29/2009 Cartesian & Polar forms: Cartesian form: z = x + jy = r cos θ + j r sin = r (cos θ + j sin ) = r e j (via Euler identity) Polar form: z = r e j r = | z |, magnitude of z = arg z = ± z , angle or argument of z Cartesian to Polar: r = | z | = x 2 + y 2 = arctan y x Polar to Cartesian: x = r cos y = r sin Example: 5 + 2j = (5 2 + 2 2 ) 1/2 e j arctan(2/5)
ME 340 notes 5/29/2009 Complex Arithmetic: z 1 = x 1 + jy 1 = r 1 e j θ 1 z 2 = x 2 + jy 2 = r 2 e j 2 z 3 = x 3 + jy 3 = r 3 e j 3 Addition: z 3 = z 1 + z 2 z 3 = z 1 + z 2 = (x 1 + jy 1 ) + (x 2 + jy 2 Example: 5 + 2j - (8 – 3 j) = -3 + 5j

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Lect08-Complex Variables - M.D Bryant ME 340 notes Complex...

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