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

Point added for a correct answer:
Points for an incorrect answer:
Ignore the questions' coefficients:

1

Calculate the integral ${\displaystyle \int _{\gamma }{\frac {1}{z}}v,dz=\int _{0}^{2\pi }{\frac {1}{\gamma (t)}}\cdot \gamma {\,}'(t)\,dt}$ with ${\displaystyle \gamma (t):=e^{i\cdot t}}$ 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).

Answer: ${\displaystyle +}$ ${\displaystyle i}$

2

Enter you the Residue ${\displaystyle res_{z_{0}}(f)}$ the Function ${\displaystyle f(z)={\frac {3x+6+i\cdot 5x}{x^{2}+2x}}}$ in the Poitt ${\displaystyle z_{0}=0}$ an. (Error tolerance 5% for the Real- and Imaginary)

Answer: ${\displaystyle +}$ ${\displaystyle i}$

3

Enter the Residue ${\displaystyle res_{z_{0}}(f)}$ of the function ${\displaystyle f(z)={\frac {3x+6+i\cdot 5x}{x^{2}+2x}}}$ at the point ${\displaystyle z_{0}=i}$. 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).

Answer: ${\displaystyle +}$ ${\displaystyle i}$

4

Enter the Residue ${\displaystyle res_{z_{0}}(f)}$ of the function ${\displaystyle f(z)={\frac {3x+6+i\cdot 5x}{x^{2}+2x}}}$ at the point ${\displaystyle z_{0}=3+5i}$. 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).

Answer: ${\displaystyle +}$ ${\displaystyle i}$

5

Enter the residue ${\displaystyle res_{z_{0}}(f)}$ of the function ${\displaystyle f(z)={\frac {3x+6+i\cdot 5x}{x^{2}+2x}}}$ at the point ${\displaystyle z_{0}=-2}$. 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).

Answer: ${\displaystyle +}$ ${\displaystyle i}$

6

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

	${\displaystyle f}$ can be locally developed into a power series for each ${\displaystyle z_{o}\in G}$.
	The partial derivatives for the real part and imaginary part of ${\displaystyle f}$ exist.
	The function is complex differentiable at a single point ${\displaystyle z_{o}}$.
	The function is complex differentiable at every point ${\displaystyle z\in G}$.
	The function is infinitely complex differentiable at every point ${\displaystyle z\in G}$.
	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.
	${\displaystyle f}$ is real differentiable, and it holds that ${\displaystyle \quad {\frac {\partial f}{\partial {\bar {z}}}}=0,}$ where ${\displaystyle {\tfrac {\partial }{\partial {\bar {z}}}}}$ is the Cauchy-Riemann operator defined by ${\displaystyle {\tfrac {\partial }{\partial {\bar {z}}}}:={\tfrac {1}{2}}\left({\tfrac {\partial }{\partial x}}+i{\tfrac {\partial }{\partial y}}\right)}$.
	The real part function ${\displaystyle f_{1}}$ and the imaginary part function ${\displaystyle f_{2}}$ with ${\displaystyle f=f_{1}+f_{2}}$ are real integrable.

See Also

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