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Imaginary Unit

Definition

Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.

𝐳:=a+bi=rexp(iφ)\boldsymbol{z} := a + b\text{i}= r \cdot \exp(\text{i}\varphi)

with Imaginary Unit i=1\text{i}= \sqrt{-1}.

Complex Plane
Complex Plane

Polarkoordinaten: 𝐳=r(cos(φ)+𝐢sin(φ))=reφ𝐢\boldsymbol{z} = r \cdot (\cos(\varphi)+\mathbf{i}\sin(\varphi))=r\cdot e^{\varphi \mathbf{i}}

r=|𝐳|=a2+b2φ=arg(𝐳)={+arccos(ar),b0arccos(ar),b<0r=|\boldsymbol{z}|=\sqrt{a^2+b^2}\varphi=\arg(\boldsymbol{z})=\begin{cases}+\arccos \left( \frac{a}{r}\right), & b \ge 0 \\ -\arccos \left( \frac{a}{r}\right), & b<0 \end{cases}

with φ[0,2π]\varphi \in [0, 2 \pi]

Rules

i2n=1n\text{i}^{2n} = -1^n    i2n+1=in\text{i}^{2n+1} = -\text{i}^n    i1=i\text{i}^{-1} = -\text{i}

Conjugate: 𝐳*=abi\boldsymbol{z}^{*} = a - b\text{i}     exp(iφ¯)=exp(iφ)\exp(\overline{i\varphi}) = \exp({-i\varphi}) 𝐳𝐳¯=|𝐳|2=a2+b2\boldsymbol{z} \overline{\boldsymbol{z}} = \left\vert{\boldsymbol{z}}\right\vert^2 = a^2+b^2

Inverse: 𝐳1=𝐳*𝐳*𝐳=abia2+b2\displaystyle \boldsymbol{z}^{-1} = \frac{\boldsymbol{z}^{*}}{\boldsymbol{z}^{*}} \boldsymbol{z}=\frac{a - b\text{i}}{a^2+b^2}

Multiplikation: 𝐳1𝐳2=r1r2(cos(φ1+φ2)+𝐢sin(φ1+φ2))\boldsymbol{z}_1\cdot \boldsymbol{z}_2=r_1r_2 ( \cos ( \varphi_1 + \varphi_2) + \mathbf{i} \sin (\varphi_1 + \varphi_2))

Division: 𝐳1𝐳2=r1r2(cos(φ1φ2)+𝐢sin(φ1φ2))\frac{\boldsymbol{z}_1}{\boldsymbol{z}_2}=\frac{r_1}{r_2} ( \cos ( \varphi_1 - \varphi_2) + \mathbf{i} \sin (\varphi_1 - \varphi_2))

n-te Potenz: 𝐳n=rnenφ𝐢=rn(cos(nφ)+𝐢sin(nφ))\boldsymbol{z}^n=r^n\cdot e^{n\varphi \mathbf{i}}= r^n (\cos (n \varphi) + \mathbf{i} \sin (n \varphi))

Logarithmus: ln(𝐳)=ln(r)+𝐢(φ+2kπ)\ln(\boldsymbol{z})=\ln(r) + \mathbf{i}(\varphi + 2k\pi) (Nicht eindeutig!)