z-complex-var-impedance

# z-complex-var-impedance - Complex numbers &amp; variables...

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Im(z) Re(z) z | z x y z x+iy = z* x-iy = z* θ -i z*=|z|e θ - θ -y e =cos(θ)+ i sin(θ) Re(z) = x Im(z) = y i= -1 2 i -1 = e =cos( ) i sin( ) i ± ± e +e cos( )= 2 i i e -e sin( )= 2 i i i i z=|z|e (x,y) 2 2 2 |z| =z z*=x + y Complex variable x 2 +1=0 want solutions to equations like x 2 =-1 Complex conjugate of z Complex z-plane note cv-im1

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3 2 i ii -i = = 4 2 2 i i i (-1)(-1) = 1 = = { } n(odd) i i; n=1,5,9. .. | -i; n= 3,7,11. .. = { } n(even) i -1; n=2,6,10. .. | 1; n= 4,8,12. .. = n= n n-0 e = (iθ) n= n= n= n n n n=0 n(even)=0 n(odd)=1 e = (iθ) (iθ) (iθ) = + 2 3 4 5 6 7 8 9 e = 1+iθ +(iθ) +(iθ) +(iθ) +(iθ) +(iθ) +(iθ) +(iθ) +(iθ) . .. { } 2 4 6 8 10 12 e = 1+(iθ) +(iθ) +(iθ) +(iθ) +(iθ) +(iθ) . .. { } 2 4 6 8 10 12 e = 1-(θ) +(θ) -(θ) +(θ) -(θ) +(θ) . .. { } 1 3 5 7 9 + (iθ) +(iθ) +(iθ) +(iθ) +(iθ) . .. { } 1 3 5 7 9 + i (θ) - (θ) + (θ) - (θ) + (θ) .

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## This note was uploaded on 04/03/2012 for the course PHY 1020 taught by Professor Kodera during the Spring '11 term at University of Florida.

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z-complex-var-impedance - Complex numbers &amp; variables...

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