Complex Number

The complex numbers are the Field $\Bbb{C}$ of numbers of the form $x+iy$, where $x$ and $y$ are Real Numbers and i is the Imaginary Number equal to $\sqrt{-1}$. When a single letter $z=x+iy$ is used to denote a complex number, it is sometimes called an ``Affix.'' The Field of complex numbers includes the Field of Real Numbers as a Subfield.

Through the Euler Formula, a complex number

$$z=x+iy$$ (1)

may be written in ``Phasor'' form

$$z = \vert z\vert(\cos \theta +i \sin \theta) = \vert z\vert e^{i\theta}.$$ (2)

Here, $\vert z\vert$ is known as the Modulus and $\theta$ is known as the Argument or Phase. The Absolute Square of $z$ is defined by $\vert z\vert^2 = zz^*$, and the argument may be computed from

$$\arg(z)=\theta=\tan^{-1}\left({y\over x}\right).$$ (3)

de Moivre's Identity relates Powers of complex numbers

$$z^n = \vert z\vert^n[\cos(n\theta)+i \sin(n\theta)].$$ (4)

Finally, the Real $\Re(z)$ and Imaginary Parts $\Im(z)$ are given by

$$\Re(z) = {\textstyle{1\over 2}}(z+z^*)$$ (5)

$$\Im(z) = {z-z^*\over 2i} = - {\textstyle{1\over 2}}i(z-z^*) = {\textstyle{1\over 2}}i(z^*-z).$$ (6)

The Powers of complex numbers can be written in closed form as follows:

$$z^n = \left[{x^n-{n\choose 2}x^{n-2}y^2+{n \choose 4}x^{n-4}y^4-\ldots}\right]$$

$$\phantom{=} + i\left[{{n\choose 1}x^{n-1}y-{n\choose 3}x^{n-3}y^3+\ldots}\right].$$ (7)

The first few are explicitly

$$z^2 = (x^2-y^2)+i(2xy)$$ (8)

$$z^3 = (x^3-3xy^2)+i(3x^2y-y^3)$$ (9)

$$z^4 = (x^4-6x^2y^2+y^4)+i(4x^3y-4xy^3)$$ (10)

$$z^5 = (x^5-10x^3y^2+5xy^4)+i(5x^4y-10x^2y^3+y^5)$$ (11)

(Abramowitz and Stegun 1972).

See also Absolute Square, Argument (Complex Number), Complex Plane, i, Imaginary Number, Modulus, Phase, Phasor, Real Number, Surreal Number

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 16-17, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 353-357, 1985.

Courant, R. and Robbins, H. ``Complex Numbers.'' §2.5 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 88-103, 1996.

Morse, P. M. and Feshbach, H. ``Complex Numbers and Variables.'' §4.1 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 349-356, 1953.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Complex Arithmetic.'' §5.4 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 171-172, 1992.