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lecture12 - Section 4.3; Section 4.4 2nd order DFQ with...

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Unformatted text preview: Section 4.3; Section 4.4 2nd order DFQ with constant coefficients, complex roots March 5, 2009 Characteristic Eqn with Complex Roots Todays Session A Summary of This Session: (1) Solve for complex roots (2) Characteristic equation with complex roots (3) Nonsynchronous solution when w n = w Characteristic Eqn with Complex Roots Eulers formula e i = cos + i sin In order to justify the above formula, we recall the MacLaurin series for e x , sin x and cos x , valid for all x : e x = 1 + x + x 2 2! + x 3 3! + x 4 4! + x 5 5! + . . . cos x = 1- x 2 2! + x 4 4!- . . . sin x = x- x 3 3! + x 5 5!- . . . So: e i x = 1 + ( i x ) + ( i x ) 2 2! + ( i x ) 3 3! + ( i x ) 4 4! + ( i x ) 5 5! + . . . Remember i 2 =- 1. Therefore: e i x = 1 + i x- x 2 2!- i x 3 3! + x 4 4! + i x 5 5! + . . . Characteristic Eqn with Complex Roots Eulers formula, contd We collect terms e i x = parenleftbigg 1- x 2 2! + x 4 4!- . . . parenrightbigg + i parenleftbigg x- x 3 3! + x 5 5!- . . . parenrightbigg Which is exactly Eulers formula. Example: (a) Calculate: e i , e i / 4 , e i / 3 , e i / 6 (b) Show that e i ( +2 k ) = e i (b) Calculate (1 + i ) 2 Answer: e i =- 1 , e i / 4 = 2 2 + i 2 2 , e i / 3 = 1 2 + i 3 2 and e i / 3 = 3 2 + i 2 ....
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lecture12 - Section 4.3; Section 4.4 2nd order DFQ with...

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