This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View Full
Document
 Spring '09
 Lotfi
 Differential Equations, Equations

Click to edit the document details