Berechnen Sie die folgenden Wurzeln: a) \(\sqrt[2]{4-2 j}\) b) \(\sqrt[3]{81 \cdot e^{-j 190^{-1}}}\) c) \(\sqrt[6]{-3+8 \mathrm{j}}\)

To solve complex number problems, it's useful to convert them into polar form. This means expressing the number in terms of its magnitude (or length) and its angle (or argument) with respect to the real axis.

For example, take the complex number \(4 - 2j\). In polar form, this is written as \(r e^{j\theta}\), where \(r\) is the magnitude and \(\theta\) is the angle.

First, find the magnitude using the formula \(\vert x + yi \vert = \sqrt{x^2 + y^2}\). For \(4 - 2j\), the magnitude is \(\sqrt{4^2 + (-2)^2} = \sqrt{20} = 2\sqrt{5}\).

Next, calculate the argument or angle with the formula \(\theta = \tan^{-1}(\frac{y}{x})\). Here, it’s \(\tan^{-1}(\frac{-2}{4}) = \tan^{-1}(-0.5) \approx -26.57^\circ\).

So, the polar form of \(4 - 2j\) is \(2\sqrt{5} e^{-j26.57^\circ}\).

De Moivre's theorem is a powerful tool for finding powers and roots of complex numbers. It states that for a complex number in polar form \(r e^{j\theta}\), its \(n\)-th root can be found by taking the \(n\)-th root of the magnitude and dividing the angle by \(n\).

For instance, to find the square root of \(4 - 2j\) with polar form \(2\sqrt{5} e^{-j26.57^\circ}\):

• The magnitude of the root is \(\sqrt{2\sqrt{5}} = \sqrt{10}\).
• The argument of the root is \(\frac{-26.57^\circ}{2} = -13.285^\circ\).

Applying de Moivre's theorem, the square root of \(4 - 2j\) is \(\sqrt{10} e^{-j13.285^\circ}\).

This theorem simplifies calculations by breaking down the process into manageable steps.

The magnitude and argument of a complex number are fundamental in converting it to polar form. The magnitude \(r\) is the distance from the complex number to the origin in the complex plane, calculated as \(r = \sqrt{x^2 + y^2}\).

For instance, the magnitude of \(-3 + 8j\) is \(\sqrt{(-3)^2 + 8^2} = \sqrt{73}\).

The argument \(\theta\) is the angle made with the positive real axis, determined using \(\theta = \tan^{-1}(\frac{y}{x})\). For \(-3 + 8j\), the angle is \(\tan^{-1}(\frac{8}{-3}) \approx 110.56^\circ\).

This polar form representation \(r e^{j\theta}\) allows for straightforward calculations involving multiplication, division, and powers or roots of complex numbers.

The complex exponential function \(e^{j\theta}\) is a key concept in working with complex numbers. It ties together Euler's formula, \(e^{j\theta} = \cos(\theta) + j\sin(\theta)\), allowing complex numbers to be expressed as exponentials.

For example, \(e^{j180^\circ} = -1\), showing how complex exponentials can represent rotations.

In polar form, the magnitude and argument of a complex number can be combined into an exponential term \(r e^{j\theta}\).

This makes operations like finding roots straightforward. Consider \( \root[3]{81 \times e^{-j0.00526}} \). Using de Moivre's theorem, we find:

• The modulus of the root is \(\root[3]{81} = 3\).
• The argument of the root is \(\frac{-0.00526}{3} = -0.00175\).

Thus, \(\root[3]{81 \times e^{-j0.00526}} = 3 e^{-j0.00175}\), simplifying complex number manipulations.