Lecture38-rev

Lecture38-rev - M AT A26 Lecture 38 1 Simple Functions of a...

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Unformatted text preview: M AT A26 Lecture 38 1 Simple Functions of a Complex Variable If P ( x ) = a + a 1 x + a 2 x 2 + ... + a n x n is a polynomial with real or complex coefficients, then the polynomial P is a perfectly well defined function function with domP = C taking values in C . Also, if f ( x ) = P ( x ) Q ( x ) where P and Q are polynomials with real or complex coefficients then the rational function f is a perfectly well defined function with domain C-{ z C | Q ( x ) = 0 } taking values in C 1.1 Examples 1. Find the domain of f ( z ) = z z 2 +1 . SOLUTION: We must determine where the denominator is zero, 1.e., we must solve z 2 + 1 = 0. Now z 2 + 1 = ( z + i )( z- i ) = 0 if and only if z = i or z =- i . Thus domf = C-{- i, i } 2. Let P ( z ) = i + (3 + 4 i ) z 2 Calculate P (2- 3 i ). Find the real and imaginary parts of P ( z ). SOLUTION: P (2- 3 i ) = i + (3 + 4 i )(2- 3 i ) 2 = i + (3 + 4 i )(4- 12 i + 9( i ) 2 ) = i + (3 + 4 i )(4- 12 i- 9) = i + (3 + 4 i )(- 5- 12 i ) = i + (- 15- 36 i- 20 i- 48( i ) 2 ) = i + (- 15- 56 i + 48) = 33- 55 i To find the real and imaginary parts of P ( z ), evaluate P ( x + iy ) P ( x + iy ) = i + (3 + 4 i )( x + iy ) 2 = i + (3 + 4 i )( x 2 + 2 ixy- y 2 ) = i + 3 x 2- 3 y 2- 8 xy + 4 x 2 i- 4 y 2 i = 3 x 2- 3 y 2- 8 xy + i (1 + 4...
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This note was uploaded on 03/04/2012 for the course MATH 10250 taught by Professor Himonas during the Fall '08 term at Notre Dame.

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Lecture38-rev - M AT A26 Lecture 38 1 Simple Functions of a...

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