User:Aleksander Stos/ComplexNumberAdvanced

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Complex numbers are defined as ordered pairs of reals:

$\mathbb {C} =\{(a,b)\colon a,b\in \mathbb {R} \}.$

Such pairs can be added and multiplied as follows

• addition: $(a,b)+(c,d)=(a+c,b+d)$
• multiplication: $(a,b)(c,d)=(ac-bd,bc+ad)$

$\scriptstyle \mathbb {C}$ with the addition and multiplication is the field of complex numbers. From another of view, $\scriptstyle \mathbb {C}$ with complex additions and multiplication by real numbers is a 2-dimesional vector space.

To perform basic computations it is convenient to introduce the imaginary unit, i=(0,1). It has the property $\scriptstyle i^{2}=-1.$ Any complex number $z=(a,b)$ can be written as $z=a+bi$ (this is often called the algebraic form) and vice-versa. The numbers a and b are called the real part and the imaginary part of z, respectively. We denote $a=\Re (z)$ and $b=\Im (z).$ Notice that i makes the multiplication quite natural:

$(a+bi)(c+di)=(ac-bd)+(bc+ad)i.$

The square root of number in the denominator in the above formula is called the modulus of z and denoted by $|z|$,

$|z|={\sqrt {a^{2}+b^{2}}}.$

We have for any two complex numbers $z_{1}$ and $z_{2}$

• $|z_{1}\cdot z_{2}|=|z_{1}|\cdot |z_{2}|;$
• $|{\frac {z_{1}}{z_{2}}}|={\frac {|z_{1}|}{|z_{2}|}},$ provided $z_{2}\not =0.$
• ${\big |}|z_{1}|-|z_{2}|{\big |}\leq |z_{1}+z_{2}|\leq |z_{1}|+|z_{2}|$

For $z=a+bi$ we define also ${\bar {z}}$, the conjugate, by ${\bar {z}}=a-bi.$ Then we have

• ${\bar {z}}_{1}\pm {\bar {z}}_{2}={\overline {(z_{1}\pm z_{2})}};$
• ${\bar {z}}_{1}\cdot {\bar {z}}_{2}={\overline {(z_{1}\cdot z_{2})}};$
• ${\overline {\left({\frac {z_{1}}{z_{2}}}\right)}}={\frac {{\bar {z}}_{1}}{{\bar {z}}_{2}}},$ provided $z_{2}\not =0;$
• $z{\bar {z}}=|z|^{2}.$

Geometric interpretation

Complex numbers may be naturally represented on the complex plane, where $z=x+iy$ corresponds to the point (x,y), see the fig. 1.

Fig. 1. Graphical representation of a complex number and its conjugate

The modulus is just the distance from the point $z=(x,y)$ and the origin. More generally, $|z_{1}-z_{2}|$ is the distance between the two given points. Furthermore, the conjugation is just the symmetry with respect to the x-axis.

Trigonometric and exponential form

As the graphical representation suggests, any complex number z=a+bi of modulus 1 (i.e. a point from the unit circle) can be written as $z=\cos \theta +i\sin \theta$ for some $\theta \in [0,2\pi ).$ So actually any (non-null) $z\in \mathbb {C}$ can be represented as

$z=r(\cos \theta +i\sin \theta ),$ where r traditionally stands for |z|.

This is the trigonometric form of the complex number z. If we adopt convention that $\theta \in [0,2\pi )$ then such $\theta$ is unique and called the argument of z. Graphically, the number $\theta$ is the (oriented) angle between the x-axis and the interval containing 0 and z. Closely related is the exponential notation. If we define complex exponential as

$e^{z}=\sum _{0}^{\infty }{\frac {z^{n}}{n!}},$

then it may be shown that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{i\theta}=\cos\theta + i\sin\theta,\quad\quad \theta\in\mathbb{R}. }

Consequently, any (non-zero) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\in \mathbb{C}} can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z= r e^{i\theta}} with the same r and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle theta} as above.

This is called the exponential form of the complex number z. It is well-adapted to perform multiplications. Indeed, for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1=r_1e^{i\theta_1}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2=r_2e^{i\theta_2}} we have

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1 z_2= r_1r_2 e^{i(\theta_1+\theta_2)}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1-\theta_2)},} provided Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2\not=0.}

The following particular case of complex multiplication is well-know as the de Moivre formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (cos\theta+i\sin\theta)^n = \cos(n\theta)+i\sin(n\theta)}

It is commonly used to linearise powers of trigonometric functions in integrals.

Fig 2. Multiplication by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} amounts to rotation by 90 degrees.

Graphically, multiplication by a constant complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=re^{i\theta}} amounts to the rotation by $\theta$ and the homothety of ratio r. In particular, the multiplication by i amounts to the rotation by the right angle (counter-clockwise), see Fig. 2.