solutions_complex

solutions_complex - i 1-i = 1 i 1-i 1 i 1 i =(1 i(1...

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SOLUTIONS TO PROBLEMS ON COMPLEX NUMBERS 1) By the quadratic formula, the roots of the equation r 2 + αr + 1 = 0 are r = - α ± α 2 - 4 2 The solutions are real if and only if α 2 - 4 0 i.e., if and only if α ≤ - 2 or α 2 2) Given the quadratic equation r 2 + iβr - 1 = 0 with real β , r = - ± p ( ) 2 + 4 2 = - ± p - β 2 + 4 2 Solutions are pure imaginary if and only if - β 2 + 4 0, i.e., if and only if β ≤ - 2 or β 2 3) 1 +
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Unformatted text preview: i 1-i = 1 + i 1-i · 1 + i 1 + i = (1 + i )(1 + i ) (1-i )(1 + i ) = 2 i 2 = i 4) c 1 e (-2+3 i ) t + c 2 e (-2 t-3 i ) t = c 1 e-2 t e (3 t ) i + c 2 e-2 t e-(3 t ) i = c 1 e-2 t (cos(3 t ) + i sin(3 t )) + c 2 e-2 t (cos(3 t )-i sin(3 t )) = ( c 1 + c 2 ) e-2 t cos(3 t ) + ( c 1-c 2 ) ie-2 t sin(3 t ) 1...
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This note was uploaded on 12/29/2010 for the course MATH 265 taught by Professor Leahkeshet during the Winter '10 term at UBC.

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