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# R01 - ENEE 241 02 READING ASSIGNMENT 1 Tue 02/03 Lecture...

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ENEE 241 02* READING ASSIGNMENT 1 Tue 02/03 Lecture Topic: complex multiplication; complex exponentials Textbook References: section 1.3 Key Points: Two complex numbers can be multiplied by expressing each number in the form z = x + jy , then using distributivity and the rule j 2 = - 1 (i.e., j is treated as the square root of - 1). The product of two complex numbers with polar coordinates ( r 1 , θ 1 ) and ( r 2 , θ 2 ) is the complex number with polar coordinates ( r 1 r 2 , θ 1 + θ 2 ). The product of a complex number z and its conjugate z * equals the square modulus | z | 2 . If z has polar coordinates ( r, θ ), its inverse z - 1 has polar coordinates ( r - 1 , - θ ). The generic complex number z = r (cos θ + j sin θ ) can be also written as z = re . As is the case with real exponentials, e j ( θ + φ ) = e e The functions cos θ and sin θ can be expressed in terms of complex exponentials: cos θ = e + e - 2 and sin θ = e - e - 2 j Theory and Examples: 1 . The most common form for a complex number z incorporates the real and imaginary parts as follows: z = x + jy This form, together with the convention that j × j = j 2 = - 1, allows us to multiply two complex numbers together. For example, (5 - 2 j )(3 - 4 j ) = 15 - 20 j - 6 j + 8 j 2 = 7 - 26 j 2 . In polar form, multiplication of complex numbers is simple. If z 1 = r 1 (cos θ 1 + j sin θ 1 ) and z 2 = r

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R01 - ENEE 241 02 READING ASSIGNMENT 1 Tue 02/03 Lecture...

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