lecture1

# lecture1 - Boise State University Department of Electrical...

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Boise State University Department of Electrical and Computer Engineering ECE212 – Circuit Analysis and Design Lecture #1: The Phasor Transform I Lecture Objectives : 1. To derive and apply Euler’s Identity 2. To derive and apply De Moivre’s Formula 3. To review complex algebra Euler’s Identity : From calculus, e is an irrational number deﬁned as e = lim n →∞ (1 + 1 n ) n = 2 . 718218 . . . n 1 10 100 1000 ··· (1 + 1 /n ) n 2.000000 2.593742 2.704814 2.716924 ··· Expanding e x as a Maclaurin/Taylor series expansion f ( x ) = f (0) + f (0) 1! x + f ′′ (0) 2! x 2 + f ′′′ (0) 3! x 3 + ··· e x = 1 + x 1! + x 2 2! + x 3 3! + x 4 4! + x 5 5! + ··· Example : e 1 = 1 + 1 1! + 1 2 2! + 1 3 3! + 1 4 4! + 1 5 5! | {z } + ··· = 2 . 716667 + ··· Replacing x by the pure imaginary number above yields e = 1 + ( ) + ( ) 2 2! + ( ) 3 3! + ( ) 4 4! + ( ) 5 5! + ··· = (1

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lecture1 - Boise State University Department of Electrical...

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