lecture12

# 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 Today’s 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 Euler’s formula e i θ = cos θ + i sin θ 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 Euler’s formula, cont’d 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 Euler’s formula. Example: (a) Calculate: e i π , e i π/ 4 , e i π/ 3 , e i π/ 6 (b) Show that e i ( θ +2 k π ) = e i θ (b) Calculate (1 + i ) 2 Answer: e i π =- 1 , e i π/ 4 = √ 2 2 + i √ 2 2 , e i π/ 3 = 1 2 + i √ 3 2 and e i π/ 3 = √ 3 2 + i 2 ....
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lecture12 - Section 4.3 Section 4.4 2nd order DFQ with...

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