/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Geben Sie zu folgenden komplexen... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 1

Geben Sie zu folgenden komplexen Zahlen die Polarkoordinatendarstellung an, $$ z_{1}=-2 \mathrm{i}, \quad z_{2}=1+\mathrm{i}, \quad z_{3}=\frac{1}{2}(-1+\sqrt{3} \mathrm{i}) $$ Zu den komplexen Zahlen mit Polarkoordinaten \(r_{4}=2\), \(\varphi_{4}=\frac{1}{2} \pi, \quad r_{5}=1, \varphi_{5}=\frac{3}{4} \pi\), bzw. \(r_{6}=3, \varphi_{6}=\frac{5}{4} \pi\) sind Real- und Imagin盲rteil gesucht.

Short Answer

Expert verified
Answer: The polar forms of the complex numbers are: \(z_1=2\operatorname{cis}\frac{3}{2}\pi\), \(z_2=\sqrt{2}\operatorname{cis}\frac{1}{4}\pi\), and \(z_3=1\operatorname{cis}\frac{2}{3}\pi\). The Cartesian forms of the complex numbers are: \(z_4=0+2i\), \(z_5=-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i\), and \(z_6=-\frac{3\sqrt{2}}{2}-\frac{3\sqrt{2}}{2}i\).

Step by step solution

01

(Step 1: Find Polar Coordinate Representation for \(z_1=-2i\))

(To find the polar form of \(z_1 = -2i,\) we first need to find the modulus and argument. The modulus is \(r_1 = \sqrt{0^2 + (-2)^2} = 2\). The argument is \(\varphi_1 = \text{atan2}(-2,0) = \frac{3}{2}\pi\). Therefore, the polar form of \(z_1= -2i\) is \(2\operatorname{cis}\frac{3}{2}\pi\).)
02

(Step 2: Find Polar Coordinate Representation for \(z_2= 1+i\))

(Similar to the previous step, we first find the modulus and argument for \(z_2 = 1+i\). Modulus: \(r_2 = \sqrt{1^2 + 1^2} = \sqrt{2}\). Argument: \(\varphi_2 = \text{atan2}(1,1) = \frac{1}{4}\pi\). So, the polar form of \(z_2=1+i\) is \(\sqrt{2}\operatorname{cis}\frac{1}{4}\pi\).)
03

(Step 3: Find Polar Coordinate Representation for \(z_3=\frac{1}{2}(-1+\sqrt{3}i)\))

(As before, we find the modulus and argument for \(z_3=\frac{1}{2}(-1+\sqrt{3}i)\). Modulus: \(r_3 = \sqrt{(-\frac{1}{2})^2 + (\frac{\sqrt{3}}{2})^2} = 1\). Argument: \(\varphi_3 = \text{atan2}(\frac{\sqrt{3}}{2}, -\frac{1}{2}) = \frac{2}{3}\pi\). Then the polar form of \(z_3=\frac{1}{2}(-1+\sqrt{3}i)\) is \(1\operatorname{cis}\frac{2}{3}\pi\).)
04

(Step 4: Find Real and Imaginary Parts for \(z_4\))

(Now, we need to find the real and imaginary parts for the given complex number in polar form \(r_4=2, \varphi_4=\frac{1}{2}\pi \). Using Euler's formula, the real part is \(x_4 = 2\cos\frac{1}{2}\pi=0\). The imaginary part is \(y_4 = 2\sin\frac{1}{2}\pi=2\). So, the complex number \(z_4\) is \(0+2i\).)
05

(Step 5: Find Real and Imaginary Parts for \(z_5\))

(Using the polar form of complex number \(r_5=1, \varphi_5=\frac{3}{4}\pi \) and Euler's formula, we find the real and imaginary parts. Real part: \(x_5 = 1\cos\frac{3}{4}\pi=-\frac{\sqrt{2}}{2}\). Imaginary part: \(y_5 = 1\sin\frac{3}{4}\pi=\frac{\sqrt{2}}{2}\). So, the complex number \(z_5\) is \(-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i\).)
06

(Step 6: Find Real and Imaginary Parts for \(z_6\))

(Finally, using the polar form of complex number \(r_6=3, \varphi_6=\frac{5}{4}\pi \) and Euler's formula, we find the real and imaginary parts for \(z_6\). Real part: \(x_6 = 3\cos\frac{5}{4}\pi=-\frac{3\sqrt{2}}{2}\). Imaginary part: \(y_6 =3\sin\frac{5}{4}\pi=-\frac{3\sqrt{2}}{2}\). So, the complex number \(z_6\) is \(-\frac{3\sqrt{2}}{2}-\frac{3\sqrt{2}}{2}i\).)

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Most popular questions from this chapter

\bullete Zeigen Sie, dass eine komplexe Zahl \(z \in \mathbb{C}\) genau dann den Betrag \(|z|=1\) hat, wenn die Identit盲t $$ \left|\frac{\bar{u} z+v}{\bar{v}_{2}+u}\right|=1 $$ f眉r alle Zahlen \(u, v \in \mathbb{C}\) mit \(|u| \neq|v|\) gilt.

\- Welche Menge von Punkten in der komplexen Ebene wird durch die Gleichung $$ M=\\{z \in \mathbb{C}|| z-3|=2| z+3 \mid\\} $$ beschrieben?

\- Berechnen Sie alle komplexen Zahlen \(z \in \mathbb{C}\), die ileichung $$ \frac{z-3}{z-i}+\frac{z-4+\mathrm{i}}{z-1}=2 \frac{-3+2 \mathrm{i}}{z^{2}-(1+\mathrm{i}) z+\mathrm{i}} $$

wendet. Dabei gibt es neben dem Neutralleiter noch drei weitere Leiter, deren Spannungen mit gleicher Frequenz und gleicher Amplitude, aber jeweils um die Phase \(2 \pi / 3\) gegeneinander verschoben sind. Demnach liegen an den unterschiedlichen Leitern die Spannungen $$ \begin{aligned} &u_{1}(t)=U_{0}(\cos (\omega t)+i \sin (\omega t)) \\ &u_{2}(t)=U_{0}\left(\cos \left(\omega t+\frac{2}{3} \pi\right)+i \sin \left(\omega t+\frac{2}{3} \pi\right)\right) \\ &u_{3}(t)=U_{0}\left(\cos \left(\omega t+\frac{4}{3} \pi\right)+i \sin \left(\omega t+\frac{4}{3} \pi\right)\right) \end{aligned} $$ an. Zeigen Sie, dass sich zu allen Zeitpunkten die Summe der Spannungen neutralisiert, d. h. $$ u_{1}(t)+u_{2}(t)+u_{3}(t)=0 $$ f眉r alle \(t \in \mathbb{R}\) gilt.

Bestimmen Sie die M枚biustransformation \(f\) mit den Abbildungseigenschaften $$ f(\mathrm{i})=0, \quad f(0)=-1, \quad f(1)=\frac{1-\mathrm{i}}{1+\mathrm{i}} $$ Wie lautet die Umkehrfunktion zu \(f ?\) Auf welche Mengen in der komplexen Zahlenebene werden die reelle Achse, d. h. \(\operatorname{Im}(z)=0\), und die obere Halbebene, d. h. \(\operatorname{Im}(z)>0\), abgebildet?
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