Complex Analysis/Quiz

Introduction

In Wikiversity, you can create a quiz to test your understanding of the learning content. The options for creating a quiz to check your knowledge are explained on the Wikiversity help page "Quiz".

Example Quiz

<quiz> { Calculate the integral $\underset{\gamma }{\int }\frac{1}{z}v,dz=\underset{0}{\overset{2\pi }{\int }}\frac{1}{\gamma (t)}\cdot \gamma {}^{\prime }(t)dt$ with $\gamma (t):=e^{i\cdot t}$ as the integration path?Enter the real part and imaginary part up to two decimal places (also e.g. 4.21 for the real and imaginary parts separately). | type="{}" } Answer: { 0.0-0.01 _6}$+${ 6.27-6.29 _6}$i$ || Enter the decimal number with a dot

{ Enter you the Residue $re{s}_{z_0}(f)$ the Function $f(z)=\frac{3x+6+i\cdot 5x}{x^2+2x}$ in the Poitt $z_0=0$ an. (Error tolerance 5% for the Real- and Imaginary) | type="{}" } Answer: { 3.0 %5 _6}$+${ 0.0 %5 _6}$i$

{ Enter the Residue $re{s}_{z_0}(f)$ of the function $f(z)=\frac{3x+6+i\cdot 5x}{x^2+2x}$ at the point $z_0=i$. Enter the values with a decimal point (e.g., 4.21) for the real and imaginary parts separately (error tolerance 5% for real and imaginary parts). | type="{}" } Answer: { 0.0 %5 _6}$+${ 0.0 %5 _6}$i$

{ Enter the Residue $re{s}_{z_0}(f)$ of the function $f(z)=\frac{3x+6+i\cdot 5x}{x^2+2x}$ at the point $z_0=3+5i$. Enter the values with a decimal point (e.g., 4.21) for the real and imaginary parts separately (error tolerance 5% for real and imaginary parts). | type="{}" } Answer: { 0.0 %5 _6}$+${ 0.0 %5 _6}$i$

{ Enter the residue $re{s}_{z_0}(f)$ of the function $f(z)=\frac{3x+6+i\cdot 5x}{x^2+2x}$ at the point $z_0=-2$. Enter the values with a decimal point (e.g., 4.21) for the real and imaginary parts separately (error tolerance 5% for real and imaginary parts). | type="{}" } Answer: { 0.0 %5 _6}$+${ 5.0 %5 _6}$i$

{ Which of the following properties are holomorphic criteria for a function $f:G\to ℂ$? }

+$f$ can be locally developed into a power series for each $z_o\in G$. -The partial derivatives for the real part and imaginary part of $f$ exist. -The function is complex differentiable at a single point $z_o$. +The function is complex differentiable at every point $z\in G$. +The function is infinitely complex differentiable at every point $z\in G$. +The real and imaginary parts satisfy the Cauchy-Riemann equations and are at least once continuously real-differentiable. +The function can be developed into a complex power series. +The function is continuous, and the line integral of the function over any closed contractible path is zero. +The function values inside a disk can be determined from the function values on the boundary using the Cauchy integral formula. +$f$ is real differentiable, and it holds that
$\frac{\partial f}{\partial \overline{z}}=0,$
where $\frac{\partial }{\partial \overline{z}}$ is the Cauchy-Riemann operator defined by $\frac{\partial }{\partial \overline{z}}:=\frac{1}{2}(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y})$. +The real part function $f_1$ and the imaginary part function $f_2$ with $f=f_1+f_2$ are real integrable. </quiz>

See Also

• Partial fraction decomposition
• Complex Analysis
• Holomorphy Criteria
• Wikiversity help page "Quiz"

Translation and Version Control

This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

• Source: Kurs:Funktionentheorie/Quiz - URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Quiz

• Date: 01/14/2024

de:Kurs:Funktionentheorie/Quiz