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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.
 Fall '11
 Wong
 Complex Numbers

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