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A Brief Review of Complex Numbers Recall the Cartesian Representation of a complex number z C : z = x + iy, x := e z, y := m z and | z | = x 2 + y 2 := r 0 . Now, since x, y, and r are real numbers, there is a unique “angle” θ [0 , 2 π ) such that cos θ = x r , sin θ = y r θ = cos - 1 ( x r ) = sin - 1 ( y r ) = tan - 1 ( y x ) . As a consequence any z also admits the Polar Representation | z | e i θ = | z | (cos θ + i sin θ ) = | z | ( x | z | + i y | z | ) = x + iy. The number θ — in radians — is called angle and is denoted by — or argument denoted by arg . Note that 0 does not have an argument. It is plain that θ is any angle from the positive real axis of the complex z -plane to the position vector representing z . Moreover, by convention, θ is positive if it is measured in the anticlockwise direction, otherwise it is negative . It follows that we can express polar representation of z C as: z = | z | e i θ = | z | z = | z | arg z, (= Re z + i Im z ) , where | z | = x 2 + y 2 := r, and θ := arg( z ) := z = tan - 1 y := Im z x := Re z . One must realize that tan θ = tan( θ + π ). This means that for each real number y x there corresponds two values for tan - 1 ( y x ) := θ [0 , 2 π ) . However since r is the magnitude of z , one ought to choose
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Unformatted text preview: θ so that r ≥ . Finally, polar representation of z is not unique , since e i θ = e i ( θ +2 k π ) , for any integer k ∈ Z . In other words, the pairs ( r, θ ) and ( r, θ + 2 k π ) are representations of the same z ∈ C . By convention one can restrict θ to (-π , π ] or to [0 , 2 π ) and refer to it as the principal value of the argument. The principle value is usually denoted by Θ or by Arg ( z ) :-π < Θ ≤ 2 π or by ≤ Θ < 2 π . We can express any argument θ in terms of its principle value Θ as θ = Θ ± 2 k π , k ∈ Z + . 1 Note that z 1 := | z 1 | ∠ z 1 and z 2 := | z 2 | ∠ z 2 ⇒ z 1 z 2 = | z 1 | | z 2 | { ∠ z 1 + ∠ z 2 } , and z 1 z 2 = | z 1 | | z 2 | { ∠ z 1-∠ z 2 } , 1 z = 1 | z | {-∠ z } . This Remark is simply a pleasant visit to the Land of Complex Numbers — in your High School backyard! 2...
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