 Complex Numbers Calculus

Complex numbers general

The complex numbers are an extension of the real numbers. Many computational rules of real numbers can be applied to complex numbers.

The theory of analytic functions dealt with functions of a complex variable.

The Origin of complex numbers is due to the release of algebraic equations. The origin of the theory of imaginary numbers, that is, all numbers whose square is a negative real number, going to the Italian mathematician Gerolamo Cardano and Rafael Bombelli in the 16th Century. The introduction of the imaginary unit i as the new number is attributed to Leonhard Euler.

Gauss plane

The complex numbers are two-dimensional and can be used as vectors in the Gaussian plane of numbers represent. On the horizontal axis (Re) of the real part and on the vertical axis is applied (Im) of the imaginary part of the complex number. Vectors may also be similar to either the complex number in Cartesian coordinates (x, y) or polar coordinates (r, φ) can be expressed. Calculator (Cartesian)

$z=x+i\phantom{\rule{0.3em}{0ex}}y$

x= y=

Calculator (Polar)

$z=r\left(\mathrm{cos}\phi +i\mathrm{sin}\phi \right)$

r= φ=

Relationships of complex conjugate numbers

For complex conjugate numbers, the following relations hold.

$\stackrel{‾}{z}=z\phantom{\rule{1.3em}{0ex}}\text{wenn}\phantom{\rule{1.3em}{0ex}}z\phantom{\rule{1.3em}{0ex}}\text{reell is;}$

$\stackrel{‾}{{z}_{1}+{z}_{2}}=\stackrel{‾}{{z}_{1}}+\stackrel{‾}{{z}_{2}}$

$\stackrel{‾}{{z}_{1}\cdot {z}_{2}}=\stackrel{‾}{{z}_{1}}\cdot \stackrel{‾}{{z}_{2}}$

$\stackrel{‾}{\left(\frac{{z}_{1}}{{z}_{2}}\right)}=\frac{\stackrel{‾}{{z}_{1}}}{{z}_{2}}$

$z\cdot \stackrel{‾}{z}={x}^{2}+{y}^{2}$

Definitions and notations for complex numbers

A complex number z consiss of a real part x and an imaginary part y. The imaginary part is characterized by the imaginary unit i.

$z=x+i\phantom{\rule{0.3em}{0ex}}y$

$\text{mit}\phantom{\rule{1em}{0ex}}x,y\phantom{\rule{0.5em}{0ex}}\in \phantom{\rule{0.5em}{0ex}}\mathbb{R}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}i=\sqrt{-1}$

The complex conjugate to z consiss of a real part x and the negative imaginary part y. This corresponds to a reflection in the real axis in the Gaussian plane.

$\stackrel{‾}{z}=x-i\phantom{\rule{0.3em}{0ex}}y$

The amount of a complex number corresponds in the Gaussian plane with the length of the vector.

$\left|z\right|=\sqrt{{x}^{2}+{y}^{2}}$

The complex number in polar coordinates.

$z=r\left(\mathrm{cos}\phi +i\mathrm{sin}\phi \right)=r{e}^{i\phi }$

$\text{with}\phantom{\rule{1em}{0ex}}r=\left|z\right|=\sqrt{{x}^{2}+{y}^{2}}$

$\text{and}\phantom{\rule{1em}{0ex}}\phi =\mathrm{atan}\frac{y}{x}$

$\text{and the Euler Formula}\phantom{\rule{1em}{0ex}}\mathrm{cos}\phi +i\mathrm{sin}\phi ={e}^{i\phi }$

Rules for computing with complex numbers

Addition and subtraction of complex numbers

The addition and subtraction of complex numbers corresponding to the addition and subtraction of the position vectors. That the real and imaginary components are added or subtracted.

$\text{With}\phantom{\rule{1.5em}{0ex}}{z}_{1}={x}_{1}+i\phantom{\rule{0.3em}{0ex}}{y}_{1}\phantom{\rule{1.5em}{0ex}}\text{and}\phantom{\rule{1.5em}{0ex}}{z}_{2}={x}_{2}+i\phantom{\rule{0.3em}{0ex}}{y}_{2}\phantom{\rule{1.5em}{0ex}}\text{is}$

${z}_{1}+{z}_{2}={x}_{1}+{x}_{2}+i\phantom{\rule{0.3em}{0ex}}\left({y}_{1}+{y}_{2}\right)$

${z}_{1}-{z}_{2}={x}_{1}-{x}_{2}+i\phantom{\rule{0.3em}{0ex}}\left({y}_{1}-{y}_{2}\right)$

Multiplication of complex numbers

The multiplication is done by the brackets, taking into account the relationship i 2 = -1 be multiplied out.

$\text{With}\phantom{\rule{1.5em}{0ex}}{z}_{1}={x}_{1}+i\phantom{\rule{0.3em}{0ex}}{y}_{1}\phantom{\rule{1.5em}{0ex}}\text{and}\phantom{\rule{1.5em}{0ex}}{z}_{2}={x}_{2}+i\phantom{\rule{0.3em}{0ex}}{y}_{2}\phantom{\rule{1.5em}{0ex}}\text{is}$

${z}_{1}\cdot {z}_{2}=\left({x}_{1}+i\phantom{\rule{0.3em}{0ex}}{y}_{1}\right)\cdot \left({x}_{2}+i\phantom{\rule{0.3em}{0ex}}{y}_{2}\right)$ $={x}_{1}\cdot {x}_{2}-{y}_{1}\cdot {y}_{2}+i\phantom{\rule{0.3em}{0ex}}\left({x}_{1}\cdot {y}_{2}+{y}_{1}\cdot {x}_{2}\right)$

The multiplication of complex numbers can also be done in trigonometric or exponential form.

$\text{With}\phantom{\rule{1.5em}{0ex}}{z}_{1}={r}_{1}\left(\mathrm{cos}\phi +i\mathrm{sin}\phi \right)={r}_{1}{e}^{i\phi }$

$\text{and}\phantom{\rule{1.5em}{0ex}}{z}_{2}={r}_{2}\left(\mathrm{cos}\psi +i\mathrm{sin}\psi \right)={r}_{2}{e}^{i\psi }\phantom{\rule{1.5em}{0ex}}\text{is}\phantom{\rule{1.5em}{0ex}}$

${z}_{1}\cdot {z}_{2}$ $={r}_{1}{r}_{2}\left(\mathrm{cos}\left(\phi +\psi \right)+i\mathrm{sin}\left(\phi +\psi \right)\right)$ $={r}_{1}{r}_{2}{e}^{i\left(\phi +\psi \right)}$

Calculator: multiplication of complex numbers

${z}_{1}={x}_{1}+i\phantom{\rule{0.3em}{0ex}}{y}_{1}$

${z}_{2}={x}_{2}+i\phantom{\rule{0.3em}{0ex}}{y}_{2}$

x1= + i y1=

x2= + i y2=

Division of complex numbers

The division is carried out by the fraction is expanded with the complex conjugate of the denominator.

$\text{With}\phantom{\rule{1.5em}{0ex}}{z}_{1}={x}_{1}+i\phantom{\rule{0.3em}{0ex}}{y}_{1}\phantom{\rule{1.5em}{0ex}}\text{and}\phantom{\rule{1.5em}{0ex}}{z}_{2}={x}_{2}+i\phantom{\rule{0.3em}{0ex}}{y}_{2}\phantom{\rule{1.5em}{0ex}}\text{is}$

$\frac{{z}_{1}}{{z}_{2}}=\frac{{x}_{1}+i\phantom{\rule{0.3em}{0ex}}{y}_{1}}{{x}_{2}+i\phantom{\rule{0.3em}{0ex}}{y}_{2}}$ $=\frac{{x}_{1}+i\phantom{\rule{0.3em}{0ex}}{y}_{1}}{{x}_{2}+i\phantom{\rule{0.3em}{0ex}}{y}_{2}}\frac{{x}_{2}-i\phantom{\rule{0.3em}{0ex}}{y}_{2}}{{x}_{2}-i\phantom{\rule{0.3em}{0ex}}{y}_{2}}$ $=\frac{{x}_{1}{x}_{2}+{y}_{1}{y}_{2}}{{x}_{2}^{2}+{y}_{2}^{2}}+i\phantom{\rule{0.3em}{0ex}}\frac{{x}_{2}{y}_{1}-{x}_{1}{y}_{2}}{{x}_{2}^{2}+{y}_{2}^{2}}$

The division of complex numbers can also be done in trigonometric or exponential form.

$\text{With}\phantom{\rule{1.5em}{0ex}}{z}_{1}={r}_{1}\left(\mathrm{cos}\phi +i\mathrm{sin}\phi \right)={r}_{1}{e}^{i\phi }$

$\text{and}\phantom{\rule{1.5em}{0ex}}{z}_{2}={r}_{2}\left(\mathrm{cos}\psi +i\mathrm{sin}\psi \right)={r}_{2}{e}^{i\psi }\phantom{\rule{1.5em}{0ex}}\text{is}\phantom{\rule{1.5em}{0ex}}$

$\frac{{z}_{1}}{{z}_{2}}$ $=\frac{{r}_{1}}{{r}_{2}}\left(\mathrm{cos}\left(\phi -\psi \right)+i\mathrm{sin}\left(\phi -\psi \right)\right)$ $=\frac{{r}_{1}}{{r}_{2}}{e}^{i\left(\phi -\psi \right)}$

Calculator: Division of complex numbers

${z}_{1}={x}_{1}+i\phantom{\rule{0.3em}{0ex}}{y}_{1}$

${z}_{2}={x}_{2}+i\phantom{\rule{0.3em}{0ex}}{y}_{2}$

x1= + i y1=

x2= + i y2=

Basic complex functions f(z)

 Complex number cartesian $z=x+i\phantom{\rule{0.3em}{0ex}}y$ Real part $\mathrm{Re\left(z\right)}=x$ Imaginary part $\mathrm{Im\left(z\right)}=y$ Konjugated complex number $\stackrel{‾}{z}=x-i\phantom{\rule{0.3em}{0ex}}y$ Amount $\left|z\right|=\sqrt{{x}^{2}+{y}^{2}}$ Argument $\mathrm{arg\left(z\right)}=\phi =\mathrm{atan}\frac{y}{x}$ Polar $z=\left|z\right|\left(\mathrm{cos}\phi +i\mathrm{sin}\phi \right)$ Quadratic ${z}^{2}={x}^{2}-{y}^{2}+i\phantom{\rule{0.3em}{0ex}}xy$ Inverse $\frac{1}{z}=\frac{x}{{x}^{2}+{y}^{2}}+i\phantom{\rule{0.3em}{0ex}}\frac{\mathrm{-y}}{{x}^{2}+{y}^{2}}$ Quadratic inverse $\frac{1}{{z}^{2}}=\frac{{x}^{2}-{y}^{2}}{{\left({x}^{2}+{y}^{2}\right)}^{2}}+i\phantom{\rule{0.3em}{0ex}}\frac{-2xy}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$ Root $\sqrt{z}=±\sqrt{\frac{x+\sqrt{{x}^{2}+{y}^{2}}}{2}}±i\phantom{\rule{0.3em}{0ex}}\sqrt{\frac{-x+\sqrt{{x}^{2}+{y}^{2}}}{2}}$ Exponential function ${e}^{z}={e}^{x}\mathrm{cos}y+i\phantom{\rule{0.3em}{0ex}}{e}^{x}\mathrm{sin}y$ Logarithm $\mathrm{ln}z=\frac{1}{2}\mathrm{ln}\left({x}^{2}+{y}^{2}\right)+i\phantom{\rule{0.3em}{0ex}}\mathrm{atan}\frac{y}{x}$ Sinus $\mathrm{sin}z=\mathrm{sin}x\mathrm{cosh}y+i\phantom{\rule{0.3em}{0ex}}\mathrm{cos}x\mathrm{sinh}y$ Cosinus $\mathrm{cos}z=\mathrm{cos}x\mathrm{cosh}y-i\phantom{\rule{0.3em}{0ex}}\mathrm{sin}x\mathrm{sinh}y$ Sinus Hyperbolicus $\mathrm{sinh}z=\mathrm{sinh}x\mathrm{cos}y+i\phantom{\rule{0.3em}{0ex}}\mathrm{cosh}x\mathrm{sin}y$ Cosinus Hyperbolicus $\mathrm{cosh}z=\mathrm{cosh}x\mathrm{cos}y-i\phantom{\rule{0.3em}{0ex}}\mathrm{sinh}x\mathrm{sin}y$ Tangens $\mathrm{tan}z=\frac{\mathrm{sin}2x}{\mathrm{cos}2x+\mathrm{cosh}2y}+i\phantom{\rule{0.3em}{0ex}}\frac{\mathrm{sinh}2y}{\mathrm{cos}2x+\mathrm{cosh}2y}$

Powers of complex numbers

The rose of a complex number in the n-th natural potency is done by using the formula of Moivre.

${z}^{n}$ $={r}^{n}\left(\mathrm{cos}n\phi +i\mathrm{sin}n\phi \right)$ $={r}^{n}{e}^{in\phi }$

$\text{with}\phantom{\rule{1em}{0ex}}r=\left|z\right|=\sqrt{{x}^{2}+{y}^{2}}$

$\text{and}\phantom{\rule{1em}{0ex}}\phi =\mathrm{atan}\frac{y}{x}$

$\text{and}\phantom{\rule{1em}{0ex}}{i}^{0}=1\text{,}\phantom{\rule{1em}{0ex}}{i}^{1}=i\text{,}\phantom{\rule{1em}{0ex}}{i}^{2}=-1\text{,}\phantom{\rule{1em}{0ex}}{i}^{3}=-i\text{...}\phantom{\rule{1em}{0ex}}$

$\text{and}\phantom{\rule{1em}{0ex}}n\in \mathbb{N}$

or with the binomial theorem

$zn =x+iyn$

$= ∑ k = 0, k gerade n n k -1 k 2 x n - k y k$

$+i ∑ k = 1, k ungerade n n k -1 k-1 2 x n - k y k$

Calculator: Complex Binomial Theorem

n=

For general complex exponents is:

${z}^{\omega }={e}^{\omega \mathrm{ln}z}$

with zω as main value. If ω is not rational so there are infinite number of solutions. 