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

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Points for an incorrect answer:
Ignore the questions' coefficients:

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

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2 Enter you the Residue $res_{z_{0}}(f)$ 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)

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3 Enter the Residue $res_{z_{0}}(f)$ 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).

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4 Enter the Residue $res_{z_{0}}(f)$ 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).

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5 Enter the residue $res_{z_{0}}(f)$ 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).

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	$f$ 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 $\quad {\frac {\partial f}{\partial {\bar {z}}}}=0,$ where ${\tfrac {\partial }{\partial {\bar {z}}}}$ is the Cauchy-Riemann operator defined by ${\tfrac {\partial }{\partial {\bar {z}}}}:={\tfrac {1}{2}}\left({\tfrac {\partial }{\partial x}}+i{\tfrac {\partial }{\partial y}}\right)$ .
	The real part function $f_{1}$ and the imaginary part function $f_{2}$ with $f=f_{1}+f_{2}$ are real integrable.

Complex Analysis/Quiz

Introduction

Example Quiz

See Also

Translation and Version Control