Intro to Analysis in-class_Part_45

Intro to Analysis in-class_Part_45 - 7.1 Complex Numbers 95...

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Unformatted text preview: 7.1 Complex Numbers 95 Definition 7.1.5. We can extend | x | to | z | : | z | := ( z z ) 1 / 2 or | x + iy | := p x 2 + y 2 . Note: | | : C R + is a continuous function. Theorem 7.1.6. 1. | z | = | z | . 2. | zw | = | z || w | . Proof. HW: show | zw | 2 = | z | 2 | w | 2 and take . 3. | Re z | | z | . Proof. a 2 a 2 + b 2 and take . 4. | z + w | | z | + | w | . Proof. | z + w | 2 = ( z + w )( z + w ) = z z + z w + zw + w w = | z | 2 + 2Re( z w ) + | w | 2 | z | 2 + 2 | z w | + | w | 2 prev = | z | 2 + 2 | z || w | + | w | 2 = ( | z | + | w | ) 2 . Note that ( x,y ) (0 , 1) = (- y,x ), so multiplying by i corresponds to rotation by 2 (ccw). In general, if | | = 1, then z corresponds to rotating z about 0. e i = X n =0 ( i ) n n ! = X k =0 ( i ) 2 k (2 k )! + X k =0 ( i ) 2 k +1 (2 k + 1)! = X k =0 (- 1) k 2 k (2 k )!...
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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Intro to Analysis in-class_Part_45 - 7.1 Complex Numbers 95...

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