Complex Algebra Review

Complex Algebra Review - Complex Algebra Review Dr. V....

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Unformatted text preview: Complex Algebra Review Dr. V. Kpuska 2/11/12 Veton Kpuska 2 Complex Algebra Elements u Definitions: u Note: Real numbers can be thought of as complex numbers with imaginary part equal to zero. C R C R then If Numbers Complex all of Set : Numbers Imaginary all of Set : Numbers Real all of Set : 1 number complex a of form Cartezian + = - jy x z x,y j 2/11/12 Veton Kpuska 3 Complex Algebra Elements { } { } z of part Imaginary z of part Real Im Re define then we If If If + = = = = = z y z x jy x z x z y jy z x R I 2/11/12 Veton Kpuska 4 Eulers Identity j e e e e j e j e j e j j j j j j j 2 cos 2 cos sin cos sin cos sin cos ---- = + = - = + = + = 2/11/12 Veton Kpuska 5 Polar Form of Complex Numbers u Magnitude of a complex number z is a generalization of the absolute value function/operator for real numbers. It is buy definition always non-negative. ( 29 z of argument) (or Angle z arg z of Magnitude radians ] ,- ( r = = + z r z r re z j R 2/11/12 Veton Kpuska 6 Polar Form of Complex Numbers u Conversion between polar and rectangular (Cartesian) forms. u For z=0+j0; called complex zero one can not define arg(0+j0). Why? ( 29 ( 29 [ ] ( 29 ( 29 ( 29 ( 29 = + = = = + = + + = + + = =- x y y x r r y r x jy x jr r jy x j r jy x re z j 1 2 2 tan sin cos sin cos sin cos 2/11/12 Veton Kpuska 7 Geometric Representation of Complex Numbers....
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Complex Algebra Review - Complex Algebra Review Dr. V....

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