complex

complex - 1 Review of complex numbers 1.1 Complex numbers...

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Unformatted text preview: 1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of- 1 to the set of real numbers: i 2 =- 1. Every complex number can be written uniquely as a + bi , where a and b are real numbers. We usually use a single letter such as z to denote the complex number a + bi . In this case a is the real part of z , written a = Re z , and b is the imaginary part of z , written b = Im z . The complex number z is real if z = Re z , or equivalently Im z = 0, and it is pure imaginary if z = (Im z ) i , or equivalently Re z = 0. In general a complex number is the sum of its real part and its imaginary part times i , and two complex numbers are equal if and only if they have the same real and imaginary parts. We add and multiply complex numbers in the obvious way: ( a 1 + b 1 i ) + ( a 2 + b 2 i ) = ( a 1 + a 2 ) + ( b 1 + b 2 ) i ; ( a 1 + b 1 i ) · ( a 2 + b 2 i ) = ( a 1 a 2- b 1 b 2 ) + ( a 1 b 2 + a 2 b 1 ) i. For example, (2 + 3 i )(- 5 + 4 i ) =- 22- 7 i. In this way, addition and multiplication are associative and commutative, multiplication distributes over addition, there is an additive identity 0 and additive inverses (- ( a + bi ) = (- a ) + (- b ) i ), and there is a multiplicative identity 1. Note also that Re( z 1 + z 2 ) = Re z 1 + Re z 2 , and similarly for the imaginary parts, but a corresponding statement does not hold for mul- tiplication. Before we discuss multiplicative inverses, let us recall complex conjugation : the complex conjugate ¯ z of a complex number z = a + bi is by definition ¯ z = a- bi . It is easy to see that: Re z = 1 2 ( z + ¯ z ); Im z = 1 2 i ( z- ¯ z ) . Thus z is real if and only if ¯ z = z and pure imaginary if and only if ¯ z =- z . More importantly, we have the following formulas which can be checked by 1 direct calculation: z 1 + z 2 = ¯ z 1 + ¯ z 2 ; z 1 · z 2 = ¯ z 1 · ¯ z 2 ; z n = (¯ z ) n ; z · ¯ z = a 2 + b 2 , where in the last line z = a + bi . Thus, z · ¯ z ≥ 0, and z · ¯ z = 0 if and only if z = 0. We set | z | = √ z · ¯ z = √ a 2 + b 2 , the length or modulus of z .For example, | 2 + i | = √ 5. Note that | z 1 | 2 | z 2 | 2 = z 1 ¯ z 1 z 2 ¯ z 2 = z 1 z 2 ¯ z 1 ¯ z 2 = ( z 1 z 2 )( z 1 z 2 ) = | z 1 z 2 | 2 , and hence | z 1 || z 2 | = | z 1 z 2 | . If z 6 = 0, then z has a multiplicative inverse: z- 1 = ¯ z | z | 2 . In terms of real and imaginary parts, this is the familiar procedure of dividing one complex number into another by “rationalizing the denominator:” if at least one of c , d is nonzero, then a + bi c + di = a + bi c + di c- di c- di = ( a + bi )( c- di ) c 2 + d 2 . Thus it is possible to divide by any nonzero complex number. For example, to express (2 + i ) / (3- 2 i ) in the form a + bi , we write 2 + i 3- 2 i = 2 + i 3- 2 i 3 + 2 i 3 + 2 i = (2 + i )(3 + 2 i ) 3 2 + 2 2 = 4 + 7 i 13 = 4 13 + 7 13 i....
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complex - 1 Review of complex numbers 1.1 Complex numbers...

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