complex - Complex Numbers: A Brief Review Any complex...

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Complex Numbers: A Brief Review Any complex number z can be written in the Cartesian form z = x + iy ,where x and y are real and i is one of the square roots of - 1 (the other being - i ). Here, x Re z is the real part of z and y Im z is its imaginary part .Re z and Im z can be interpreted as the x and y components of a point representing z on the complex plane. By comparing Taylor series expansions, one can verify Euler’s formula: e =cos θ + i sin θ for any real θ (expressed in radians). This leads to an alternative polar form z = re for any complex number in terms of two reals: r ≡| z |≥ 0isthe magnitude or modulus of z ,and θ arg z is its argument or phase . We can convert between Cartesian and polar forms for complex number z using r = p x 2 + y 2 x = r cos θ θ =atan( y/x ) y = r sin θ Arguments θ and θ +2 πn for integer n describe the same point on the complex plane. The principal argument Arg z is defined to satisfy - π< Arg z π . r x y z z Re Im z θ The real and imaginary parts of complex numbers are added/subtracted separately: z 1 ± z 2 =( x 1 + iy 1 ) ± ( x 2 + iy 2 )=( x 1 ± x 2 )+ i ( y 1 ± y 2 ) . Complex numbers are multiplied/divided by multiplying/dividing their magnitudes and adding/subtracting their arguments:
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