Free Printable Analytic Complex Function Quiz

1.

What are the Cauchy-Riemann equations and how are they used to test for analyticity of a complex function?

a)

The Cauchy-Riemann equations are only applicable to linear functions

b)

The Cauchy-Riemann equations are used to test for the continuity of a complex function

c)

The Cauchy-Riemann equations are a pair of partial differential equations that must be satisfied by a complex function in order for it to be analytic. They are used to test for the analyticity of a complex function by checking if the function satisfies these equations.

d)

The Cauchy-Riemann equations are used to solve for real numbers in a complex function

2.

Define harmonic functions in the context of complex analysis and explain their significance.

a)

Functions that satisfy Laplace's equation and are significant in various physical and mathematical contexts.

b)

Functions that satisfy Laplace's equation and are insignificant in various physical and mathematical contexts.

c)

Functions that satisfy Euler's equation and are insignificant in various physical and mathematical contexts.

d)

Functions that satisfy Laplace's equation and are significant only in mathematical contexts.

3.

Find the derivative of the complex function $f(z)=z^2+2iz-1$ .

a)

$f'(z)=2z+2i$

b)

$f'(z)=4z+2i$

c)

$f'(z)=z^2+2i$

d)

$f'(z)=2z-2i$

4.

Calculate the limit of the complex function $g(z)=\frac{(z^2-1)}{(z-1)}$ as z approaches 1.

a)

2

b)

0

c)

3

d)

5

5.

How do the Cauchy-Riemann equations relate to the concept of holomorphic functions?

a)

They provide a way to test if a function is differentiable at a point in the complex plane.

b)

They are only applicable to linear functions

c)

They are used to calculate real numbers in a complex function

d)

They are used to determine the absolute value of a complex number

6.

What are some properties of harmonic functions that make them useful in complex analysis?

a)

Infinitely differentiable and satisfy Laplace's equation

b)

Can only be differentiated once

c)

Always have a real part greater than zero

d)

Satisfy Laplace's equation but not Cauchy-Riemann equations

7.

Find the derivative of the complex function $h(z)=e^z+z^2-1$ .

a)

$h'(z)=e^z-2z$

b)

$h'(z)=e^z-z^2-1$

c)

$h'(z)=e^z+2z$

d)

$h'(z)=e^z+z^2$

8.

Evaluate $\lim_{z\rightarrow-2i}=\frac{(z+2i)}{(z^4-16)}$ .

a)

$\frac{i^2}{32}$

b)

$0$

c)

$i$

d)

$\frac{-i}{32}$

9.

Find the derivative of the complex function $g(z)=e^z*\sin(z)$ .

a)

$g'(z)=e^z*\cos(z)$

b)

$g'(z)=e^z*\sin(z)$

c)

$g'(z)=e^z*\cos(z)+e^z*\sin(z)$

d)

$g'(z)=e^z*\cos(z)-e^z*\sin(z)$

10.

Find the derivative of the complex function $f(z)=e^{\left(iz\right)}+z^3-2i$ .

a)

$f'(z)=i*e^{\left(iz\right)}+3z^2$

b)

$f'(z)=-i*e^{\left(iz\right)}+3z^2$

c)

$f'(z)=i*e^{\left(iz\right)}-3z^2$

d)

$f'(z)=-i*e^{\left(iz\right)}-3z^2$

11.

Calculate the limit of the complex function $f(z)=\frac{(z^3-1)}{\left(z-1\right)}$ as z approaches 1.

a)

3

b)

0

c)

1

d)

2

12.

True or False: If $u\left(z\right)$ is harmonic and $f\left(z\right)$ is analytic in a region $\Omega$ , then $u\left(\overline{z}\right)$ is analytic and $\overline{f\left(\overline{z}\right)}$ is harmonic.

a)

True

b)

False

13.

Consider the function f(z)=z² . Which of the following are true?

a)

f is conformal at all points

b)

f is differentiable

c)

f is analytic

d)

f is continuous

14.

Check whether the function $u=x^3−3xy^2+3x^2−3y^2+1$ is harmonic.

4 lines

15.

If $f(z)=u(x,y)+iv(x,y)$ is analytic on a region A then both u and v are harmonic functions on A.

a)

True

b)

False

Analytic Complex Function Quiz

Total questions: 15

Worksheet time: 25mins

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