Master Complex Differentiability with Cauchy-Riemann Equations

Question 1

Which of the following complex functions satisfies the Cauchy-Riemann equations at the origin (0,0)?

Accepted Answer

z^2 + 2z

Answer

z^2

Answer

z^2 + z^3

Answer

z^3

Question 2

What is the boundary of the following set: {z | |z| < 1 and |z - 2| > 1}?

Accepted Answer

Two circles: unit circle and circle centered at 2 with radius 1

Answer

Unit circle

Answer

Circle centered at 2 with radius 1

Answer

Empty set

Question 3

Which of the following is a branch of the natural logarithm function?

Accepted Answer

Log(z) = ln(|z|) + i arg(z)

Answer

Log(z) = ln(|z|) - i arg(z)

Answer

Log(z) = |z| + i arg(z)

Answer

Log(z) = |z| - i arg(z)

Question 4

What is the residue of the function f(z) = 1/(z - 1) at z = 1?

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 5

Consider the complex function f(z) = z^2. At the point z = 1 + i, what is the geometric interpretation of the Cauchy-Riemann equations?

Accepted Answer

The tangent to the level curve of f(z) is perpendicular to the tangent to the coordinate curve.

Answer

The tangent to the level curve of f(z) is parallel to the tangent to the coordinate curve.

Answer

The level curve of f(z) is a circle centered at z = 1 + i.

Answer

The coordinate curve is a circle centered at z = 1 + i.

Question 6

Which of the following is a sufficient condition for a complex function f(z) to be differentiable at z0?

Accepted Answer

The Cauchy-Riemann equations hold at z0.

Answer

The partial derivatives of f(z) exist and are continuous at z0.

Answer

The limit of f(z) exists as z approaches z0.

Answer

The function f(z) is continuous at z0.

Question 7

Which of the following functions is NOT holomorphic?

Accepted Answer

f(z) = |z|^2

Answer

f(z) = z^2

Answer

f(z) = e^z

Answer

f(z) = 1/(z^2 - 1)

Question 8

If u and v are the real and imaginary parts of a holomorphic function f(z), respectively, which of the following equations holds?

Accepted Answer

u_x = v_y and u_y = -v_x

Answer

u_x = -v_y and u_y = v_x

Answer

u_x = v_x and u_y = -v_y

Answer

u_x = -v_x and u_y = v_y

Question 9

Which of the following is the complex conjugate of z = x + iy?

Accepted Answer

z* = x - iy

Answer

z* = -x + iy

Answer

z* = -x - iy

Answer

z* = x + iy

Question 10

Is the complex function f(z) = |z|^2 + 2z analytic?

Accepted Answer

No

Answer

Yes

Question 11

Which of the following statements accurately describes the relationship between the Cauchy-Riemann Equations and complex differentiability?

Accepted Answer

The Cauchy-Riemann Equations are both necessary and sufficient conditions for complex differentiability.

Answer

The Cauchy-Riemann Equations are necessary but not sufficient conditions for complex differentiability.

Answer

The Cauchy-Riemann Equations are sufficient but not necessary conditions for complex differentiability.

Question 12

Which of the following is a necessary and sufficient condition for a complex function f(z) to be differentiable at a point z0?

Accepted Answer

f(z) satisfies the Cauchy-Riemann equations at z0.

Answer

f(z) is continuous at z0.

Answer

f(z) has a derivative at z0.

Answer

f(z) is holomorphic at z0.

Question 13

Consider the function f(z) = z^2. Calculate its partial derivatives at z = 1 + i.

Accepted Answer

∂f/∂x = 2 + 2i, ∂f/∂y = 2 - 2i

Answer

Partial derivatives do not exist at z = 1 + i

Answer

∂f/∂x = 2, ∂f/∂y = 2i

Answer

∂f/∂x = 1 + i, ∂f/∂y = 1 - i

Question 14

Which of the following regions is guaranteed to contain a non-constant holomorphic function?

Accepted Answer

Simply connected region

Answer

Multiply connected region

Answer

Region with singularities

Answer

Region with boundary

Question 15

Consider the function f(z) = e^z. Verify its holomorphicity using the Cauchy-Riemann equations.

Accepted Answer

∂f/∂x = e^z, ∂f/∂y = ie^z

Answer

∂f/∂x = 0, ∂f/∂y = e^z

Answer

∂f/∂x = e^z, ∂f/∂y = -ie^z

Answer

∂f/∂x = ie^z, ∂f/∂y = e^z

Question 16

Determine the order of the pole of the function f(z) = 1/(z-2)^3 at z = 2.

Accepted Answer

3

Answer

2

Answer

1

Question 17

Which of the following is the correct Cauchy-Riemann equation for a complex function w = f(z)?

Accepted Answer

∂u/∂x = ∂v/∂y

Answer

∂u/∂y = ∂v/∂x

Answer

∂w/∂z = 0

Answer

∂z/∂w = 0

Question 18

A complex function f(z) is holomorphic if and only if it satisfies the Cauchy-Riemann equations at:

Accepted Answer

Every point in its domain

Answer

Interior points of its domain

Answer

Boundary points of its domain

Answer

A single point in its domain

Question 19

Which of the following is a necessary condition for a complex function to be differentiable at a point z?

Accepted Answer

It satisfies the Cauchy-Riemann equations at z.

Answer

It has a Taylor series expansion at z.

Answer

It is continuous at z.

Question 20

The Cauchy-Riemann equations are primarily used to:

Accepted Answer

Characterize complex differentiability.

Answer

Solve Laplace's equation.

Answer

Determine the convergence of infinite series.

Answer

Find the roots of polynomial equations.

Question 21

If f(z) is analytic in a domain and has a non-zero derivative at a point z0 in that domain, then:

Accepted Answer

f(z) is conformal at z0.

Answer

f(z) is harmonic at z0.

Answer

f(z) is entire.

Answer

f(z) is bounded in the domain.

Question 22

Which of the following is NOT a consequence of the Cauchy-Riemann equations?

Accepted Answer

Holomorphic functions have constant modulus.

Answer

Holomorphic functions have zero divergence.

Answer

Holomorphic functions are analytic.

Answer

Holomorphic functions are path-independent.

Question 23

Which of the following is a sufficient condition for a complex function to be holomorphic in a domain?

Accepted Answer

Satisfies the Cauchy-Riemann Equations throughout the domain

Answer

Continuous in the domain

Answer

Bounded in the domain

Question 24

If u(x,y) = x^2 - y^2 and v(x,y) = 2xy, determine if the function f(z) = u(x,y) + iv(x,y) is analytic.

Accepted Answer

Analytic

Answer

Not analytic

Answer

Meromorphic

Answer

Holomorphic

Question 25

What is the relationship between the Cauchy-Riemann Equations and the complex derivative?

Accepted Answer

The Cauchy-Riemann Equations are necessary and sufficient conditions for the existence of the complex derivative.

Answer

The Cauchy-Riemann Equations are necessary but not sufficient conditions for the existence of the complex derivative.

Question 26

Which of the following is a consequence of the Cauchy-Riemann Equations?

Accepted Answer

The level curves of u(x,y) and v(x,y) are orthogonal to each other.

Answer

The level curves of u(x,y) and v(x,y) are circles.

Answer

The level curves of u(x,y) and v(x,y) are parallel to each other.

Question 27

What is the significance of the Cauchy-Riemann Equations in complex analysis?

Accepted Answer

They provide necessary and sufficient conditions for complex differentiability and holomorphy.

Answer

They provide sufficient but not necessary conditions for complex differentiability.

Answer

They provide necessary but not sufficient conditions for complex differentiability.

Question 28

Which of the following is a necessary and sufficient condition for a complex function (f(z)) to be differentiable at (z = z_0)?

Accepted Answer

Satisfies the Cauchy-Riemann equations at (z = z_0)

Answer

Continuity of (f(z)) at (z = z_0)

Answer

Existence of the limit of (f(z)) as (z) approaches (z_0)

Question 29

Which of the following functions satisfies the Cauchy-Riemann equations and is therefore holomorphic?

Accepted Answer

f(z) = e^z

Answer

f(z) = z*

Answer

f(z) = |z|^2

Answer

f(z) = arg(z)

Question 30

If (f(z)) is holomorphic, then its real and imaginary parts, (u(x, y)) and (v(x, y)) respectively, satisfy the Laplace equation:

Accepted Answer

(u_{xx} + u_{yy} = 0) and (v_{xx} + v_{yy} = 0)

Answer

(u_{xx} + v_{xx} = 0) and (u_{yy} + v_{yy} = 0)

Answer

(u_{xy} + v_{xy} = 0)

Answer

(u_{xx} - u_{yy} = 0) and (v_{xx} - v_{yy} = 0)

Question 31

Which of the following complex functions has a non-zero derivative at (z = 0)?

Accepted Answer

f(z) = z^2

Answer

f(z) = |z|^2

Answer

f(z) = arg(z)

Answer

f(z) = z*

Question 32

If a complex function (f(z)) is continuous on a closed contour, the integral of (f(z)) around that contour is zero if and only if:

Accepted Answer

f(z) is holomorphic inside the contour.

Answer

f(z) is differentiable inside the contour.

Answer

f(z) is continuous inside the contour.

Question 33

The Cauchy-Riemann equations are closely related to the concept of:

Accepted Answer

Conformal mappings.

Answer

Riemann surfaces.

Answer

Harmonic functions.

Answer

Elliptic integrals.

Question 34

Let f(z) = u(x, y) + iv(x, y) be a complex function. Which of the following is the Cauchy-Riemann equation for differentiability at a point (z_0)?

Accepted Answer

$rac{∂u}{∂x} = rac{∂v}{∂y}$ and $rac{∂u}{∂y} = -rac{∂v}{∂x}$

Answer

$rac{∂^2u}{∂x∂y} = rac{∂^2v}{∂x∂y}$

Answer

$rac{∂u}{∂x} = -rac{∂v}{∂y}$ and $rac{∂u}{∂y} = rac{∂v}{∂x}$

Answer

$rac{∂u}{∂x} = rac{∂v}{∂x}$ and $rac{∂u}{∂y} = rac{∂v}{∂y}$

Question 35

If a function *f(z)* is holomorphic, what can we definitively say about its derivative *f'(z)*?

Accepted Answer

The derivative *f'(z)* exists and is continuous at every point in the domain of *f(z)*.

Answer

The derivative *f'(z)* may not exist at any point in the domain of *f(z)*.

Answer

The derivative *f'(z)* exists, but its continuity is not guaranteed.

Question 36

Which of the following functions is NOT holomorphic in the entire complex plane?

Accepted Answer

f(z) = 1/z

Answer

f(z) = sin(z)

Answer

f(z) = e^z

Answer

f(z) = z^2 + 1

Question 37

Given a complex function *f(z) = u(x, y) + iv(x, y)*, where *u(x, y) = x^2 - y^2* and *v(x, y) = 2xy*, is *f(z)* holomorphic?

Accepted Answer

Yes, because it satisfies the Cauchy-Riemann equations.

Answer

No, because it does not satisfy the Cauchy-Riemann equations.

Answer

Yes, because it is continuous and differentiable.

Question 38

What is a direct consequence of the Cauchy-Riemann equations when applied to a complex function *f(z) = u(x, y) + iv(x, y)*?

Accepted Answer

The level curves of *u(x, y)* and *v(x, y)* intersect orthogonally at every point.

Answer

The function *f(z)* is continuous at every point in its domain.

Answer

The function *f(z)* is differentiable at every point in its domain.

Question 39

Which of the following is NOT a necessary condition for a complex function to be holomorphic?

Accepted Answer

Continuity

Answer

Analyticity

Answer

Satisfying the Cauchy-Riemann equations

Answer

Differentiability

Question 40

What is a sufficient condition for a complex function *f(z) = u(x, y) + iv(x, y)* to be holomorphic?

Accepted Answer

*u(x, y)* and *v(x, y)* have continuous first-order partial derivatives and satisfy the Cauchy-Riemann equations.

Answer

*u(x, y)* and *v(x, y)* have continuous second-order partial derivatives.

Question 41

If a complex function *f(z)* is differentiable at a point *z0*, which of the following must be true?

Accepted Answer

The derivative *f'(z0)* exists and is continuous at *z0*

Answer

The derivative *f'(z0)* exists and is differentiable at *z0*

Answer

The derivative *f'(z0)* exists, but may not be continuous at *z0*

Question 42

Which of the following is NOT a direct consequence of the Cauchy-Riemann equations for a complex function *f(z)*?

Accepted Answer

The derivative of *f(z)* is continuous.

Answer

*f(z)* is a conformal mapping.

Answer

*f(z)* is a harmonic function.

Answer

*f(z)* is an analytic function.

Question 43

Which of the following is a sufficient condition for a complex function to be holomorphic?

Accepted Answer

The function satisfies the Cauchy-Riemann equations and is continuous.

Answer

The function is continuous and has a continuous derivative.

Answer

The function is continuous and has a bounded derivative.

Question 44

If a function *f(z)* is holomorphic, which of the following must be true about its real and imaginary parts, *u(x, y)* and *v(x, y)*, respectively?

Accepted Answer

Both *u(x, y)* and *v(x, y)* satisfy Laplace's equation.

Answer

Neither *u(x, y)* nor *v(x, y)* satisfies Laplace's equation.

Answer

Only *v(x, y)* satisfies Laplace's equation.

Answer

Only *u(x, y)* satisfies Laplace's equation.

Question 45

Which of the following domains is a simply connected domain?

Accepted Answer

A disk

Answer

A rectangle with a hole

Answer

A figure-eight shape

Answer

An annulus (a ring-shaped region)

Question 46

Which of the following functions has a removable singularity at *z = 0*?

Accepted Answer

f(z) = (z^2 - 1)/(z - 1)

Answer

f(z) = e^z

Answer

f(z) = 1/(z - 1)

Answer

f(z) = 1/z

Question 47

What are the Cauchy-Riemann equations for a complex function *f(z) = u(x, y) + iv(x, y)*?

Accepted Answer

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad 	ext{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$

Answer

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad 	ext{and} \quad \frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}$$

Question 48

If a complex function *f(z) = u(x, y) + iv(x, y)* is complex differentiable, which of the following is true?

Accepted Answer

Both *u(x, y)* and *v(x, y)* satisfy Laplace's equation.

Answer

Only *v(x, y)* satisfies Laplace's equation.

Answer

Neither *u(x, y)* nor *v(x, y)* satisfies Laplace's equation.

Answer

Only *u(x, y)* satisfies Laplace's equation.

Question 49

Let *f(z)* be a holomorphic function. Which of the following statements is true?

Accepted Answer

The real and imaginary parts of *f(z)* are harmonic functions.

Answer

The real part of *f(z)* is harmonic, but the imaginary part is not.

Answer

The imaginary part of *f(z)* is harmonic, but the real part is not.

Question 50

Which of the following functions satisfies the Cauchy-Riemann equations?

Accepted Answer

f(z) = z^2

Answer

f(z) =  overline{z}

Answer

f(z) = |z|

Answer

f(z) = arg(z)

Question 51

Which of the following is a necessary condition for a complex function f(z) to be differentiable at z = z_0?

Accepted Answer

Both partial derivatives of the real part u(x, y) of f(z) exist and are continuous at (x_0, y_0).

Answer

The partial derivative of u(x, y) with respect to y exists at (x_0, y_0).

Answer

The partial derivative of u(x, y) with respect to x exists at (x_0, y_0).

Question 52

Which of the following is a sufficient condition for a complex function f(z) to be holomorphic?

Accepted Answer

The Cauchy-Riemann equations are satisfied at every point in some open set containing z_0.

Answer

The Cauchy-Riemann equations are satisfied at z_0.

Answer

The partial derivatives of f(z) exist everywhere in some open set containing z_0.

Question 53

Which of the following functions is NOT holomorphic?

Accepted Answer

|z|^2

Answer

e^z

Answer

z^3

Answer

sin(z)

Question 54

Let f(z) be a holomorphic function and C be a closed contour in the complex plane. Which of the following integrals is guaranteed to be zero?

Accepted Answer

∮_C f(z) dz

Answer

∫_C overline{f(z)} dz

Answer

∫_C f(z) dz

Answer

∮_C overline{f(z)} dz

Question 55

Which of the following is a consequence of the Cauchy-Riemann equations for a holomorphic function f(z)?

Accepted Answer

The level curves of the real and imaginary parts of f(z) are perpendicular at every point.

Answer

The level curves of the real and imaginary parts of f(z) are parallel at every point.

Answer

The level curves of the real part of f(z) are circles.

Question 56

Consider a complex function f(z) with u(x, y) = x^2 - y^2 and v(x, y) = 2xy. Is f(z) holomorphic?

Accepted Answer

Yes, because it satisfies the Cauchy-Riemann equations.

Answer

No, because it does not satisfy the Cauchy-Riemann equations.

Answer

Yes, because it is continuous.