Master Complex Integration: Enhance Your Mathematical Skills

Question 1

Find the value of the contour integral ∮_C (z+2)/(z−i) dz, where C is the circle centered at the origin with radius 2, traversed counterclockwise.

Accepted Answer

2πi

Answer

0

Answer

-2πi

Question 2

Using the Cauchy integral formula, what is the value of the contour integral of a holomorphic function f(z) around a positively oriented simple closed curve C?

Accepted Answer

2πi times the sum of the residues of f(z) at all poles inside C

Answer

2πi times the sum of the values of f(z) at all poles inside C

Answer

The area of the region enclosed by C

Answer

0

Question 3

Which of the following is a valid closed path for integration in complex analysis?

Accepted Answer

A circle

Answer

A straight line segment

Answer

A parabola

Answer

A sine curve

Question 4

Which of the following is an example of a real integral that can be evaluated using the method of residues?

Accepted Answer

∫[0,∞] sin(x)/x dx

Answer

∫[-π,π] cos(x) dx

Answer

∫[0,1] x^2 dx

Answer

∫[0,2π] e^x dx

Question 5

Which of the following correctly states Cauchy's integral formula?

Accepted Answer

f(z) = (1/2πi) ∫(f(w)/(w - z)) dw, where the integral is taken counterclockwise around a circle enclosing z.

Answer

f(z) = (1/2πi) ∫(f(w)/(z - w)) dw, where the integral is taken clockwise around a circle enclosing z.

Answer

f(z) = (1/2π) ∫(f(w)/(w - z)) dw, where the integral is taken counterclockwise around a circle enclosing z.

Answer

f(z) = (1/2π) ∫(f(w)/(z - w)) dw, where the integral is taken clockwise around a circle enclosing z.

Question 6

What is the order of the pole of the function f(z) = 1/(z - 2)^4 at z = 2?

Accepted Answer

4

Answer

1

Answer

2

Answer

3

Question 7

What is the contour integral of f(z) = 1/(z - a) around the circle |z - a| = R, counterclockwise?

Accepted Answer

2πi

Answer

0

Answer

-2πi

Answer

∞

Question 8

Which of the following is the formula for the complex integral of a function f(z) along a contour C?

Accepted Answer

∫f(z) dz

Answer

∫f(z) dC

Answer

∫f(Re(z)) dz

Answer

∫f(Im(z)) dz

Question 9

What is the value of the integral ∫z^2 dz, where C is the circle |z| = 1 traversed counterclockwise?

Accepted Answer

2πi

Answer

0

Answer

-2πi

Answer

πi

Question 10

What is the value of the integral ∫0^∞ 1/(x^2+1) dx using the residue theorem?

Accepted Answer

π/2

Answer

π

Answer

2π

Answer

π/4

Question 11

Which of the following is NOT a property of analytic functions?

Accepted Answer

They can have removable singularities.

Answer

They have derivatives of all orders.

Answer

They are continuous on their domain.

Answer

They map open sets to open sets.

Question 12

What is the order of the pole of the function f(z) = 1/(z-a)^3 at z = a?

Accepted Answer

3

Answer

1

Answer

2

Answer

∞

Question 13

Which of the following functions is not entire?

Accepted Answer

1/(z-a)

Answer

e^z

Answer

sin(z)

Answer

z^2

Question 14

What is the value of the integral ∫e^z dz, where C is the unit circle centered at the origin traversed clockwise?

Accepted Answer

-2πi

Answer

0

Answer

2πi

Answer

πi

Question 15

Evaluate the line integral ∫_C (z² + 1) dz, where C is the circle |z| = 2 traversed counterclockwise.

Accepted Answer

2πi

Answer

0

Answer

-2πi

Answer

4πi

Question 16

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

Accepted Answer

1

Answer

0

Answer

-1

Answer

∞

Question 17

What is the formula for the Taylor series of a function f(z) analytic at z = a?

Accepted Answer

f(z) = Σ_(n=0)^∞ a_n (z - a)^n

Answer

f(z) = Σ_(n=1)^∞ a_n (z - a)^n

Answer

f(z) = Σ_(n=0)^∞ a_n z^n

Answer

f(z) = Σ_(n=1)^∞ a_n z^n

Question 18

For a complex function f(z) to be analytic at a point z0, which condition must be met?

Accepted Answer

f(z) has a continuous complex derivative at z0.

Answer

f(z) has a continuous second complex derivative at z0.

Answer

f(z) is continuous at z0.

Question 19

Which of the following is a residue of the function f(z) = (z - 2)/((z-1)(z+2)) at z = 2?

Accepted Answer

1/2

Answer

1

Answer

-1/2

Answer

0

Question 20

Which of the following is a necessary condition for a function to be analytic?

Accepted Answer

It must satisfy the Cauchy-Riemann equations at every point in its domain.

Answer

It must be continuous on its domain.

Answer

Its derivative must exist at every point in its domain.

Answer

It must be bounded.

Question 21

How can the residue theorem be applied to calculate the definite integral of a periodic function over a closed interval?

Accepted Answer

Integrating the function along a closed contour enclosing the interval

Answer

Calculating the residue at infinity

Answer

Using the Cauchy integral formula to evaluate the integral at a point on the interval

Question 22

Which of the following is NOT a valid contour used in Cauchy's Integral Formula?

Accepted Answer

A closed curve that does not enclose the point of interest

Answer

A curve that starts and ends at the same point

Answer

A curve that does not intersect itself

Question 23

What is a key implication of the Residue Theorem?

Accepted Answer

The integral of a meromorphic function over a closed contour is equal to 2πi times the sum of the residues at its poles inside the contour.

Answer

The integral of a holomorphic function over a closed contour is always zero.

Question 24

What is the order of the pole of the function f(z) = 1 / (z - a)^n at z = a?

Accepted Answer

n

Answer

1

Answer

n-1

Answer

∞

Question 25

Find the residue of f(z) = 1/(z - a) at z = a, where a is a complex constant.

Accepted Answer

1

Answer

0

Answer

-a

Answer

a

Question 26

Which contour integral is equal to the real integral ∫_0^∞ 1/(1 + x^2) dx?

Accepted Answer

∮_C 1/(z^2 + 1) dz

Answer

∮_C 1/(z^2 - 1) dz

Answer

∮_C 1/z dz

Question 27

Evaluate the contour integral ∮_C z/(z^2 + 1) dz, where C is a closed path that encloses the origin.

Accepted Answer

2πi

Answer

πi

Answer

0

Answer

-2πi

Question 28

If f(z) is an analytic function on a simply connected domain and f(z) = f(w) for all z and w in the domain, what can be concluded about f(z)?

Accepted Answer

f(z) is a constant function.

Answer

f(z) is a polynomial function.

Answer

f(z) is a rational function.

Answer

f(z) is a transcendental function.

Question 29

If f(z) is continuous on a positively oriented simple closed curve C and analytic inside C, what is the value of the integral ∮_C f(z) dz?

Accepted Answer

0

Answer

2πi

Answer

-πi

Answer

πi

Question 30

Which of the following is a necessary condition for a function f(z) to have a Laurent series expansion at z = a?

Accepted Answer

f(z) must be analytic in a punctured neighborhood of z = a.

Answer

f(z) must be continuous in a neighborhood of z = a.

Answer

f(z) must have a pole at z = a.

Question 31

Which of the following integrals can be evaluated using the residue theorem?

Accepted Answer

∫[0, 2π] (1 + cos θ) / (2 - cos θ) dθ

Answer

∫[0, π/2] sin(x) / x dx

Answer

∫[0, ∞] e^(-x^2) dx

Answer

∫[0, 1] x / (1 - x^2) dx

Question 32

Evaluate the real integral ∫[0, 2π] sin(θ) / (2 + cos θ) dθ using the residue theorem.

Accepted Answer

π

Answer

0

Answer

π/2

Answer

2π

Question 33

Which of the following statements is true about the function f(z) = 1 / (z^2 + 1)?

Accepted Answer

It has two simple poles at z = ±i.

Answer

It has a branch cut along the imaginary axis.

Answer

It has no poles.

Answer

It has a single pole at z = 0.

Question 34

Which of the following functions has a branch cut along the negative real axis?

Accepted Answer

f(z) = √(z^2 + 1)

Answer

f(z) = z^2

Answer

f(z) = e^z

Answer

f(z) = sin(z)

Question 35

Which formula represents Cauchy's integral formula?

Accepted Answer

f(z0) = (1/2πi) ∫[γ] f(z) / (z - z0) dz

Answer

f(z) = (1/2πi) ∫[γ] f(z0) / (z - z0) dz

Answer

f'(z) = (1/2πi) ∫[γ] f(z) / (z - z0)^2 dz

Answer

f(z0) = (1/2π) ∫[γ] f(z) / (z - z0) dz

Question 36

Determine the value of the line integral of the function $f(z) = z^2$ along the line segment connecting $z = 0$ and $z = 1 + i$.

Accepted Answer

$\frac{2}{3} + \frac{2}{3}i$

Answer

$1 + i$

Answer

$\frac{1}{2} + \frac{1}{2}i$

Answer

$0$

Question 37

For a function $f(z)$ that is analytic inside and on a simple closed contour $C$, and a point $z_0$ within $C$, determine the formula for calculating the derivative of $f(z)$ at $z_0$.

Accepted Answer

$\frac{1}{2\pi i} \int_C \frac{f(z)}{(z - z_0)^2} dz$

Answer

$\frac{1}{2\pi i} \int_C \frac{f(z)}{z - z_0} dz$

Answer

$\int_C \frac{f(z)}{(z - z_0)^2} dz$

Question 38

Find the residue of the function $f(z) = \frac{z}{sin(z)}$ at the point $z = \pi$.

Accepted Answer

$-\pi$

Answer

$1$

Answer

$0$

Answer

$\pi$

Question 39

Use the residue theorem to evaluate the integral $\int_C \frac{e^z}{z(z-1)^2} dz$, where $C$ is the circle centered at the origin with radius 2.

Accepted Answer

$2 \pi i (e - 1)$

Answer

$\pi i (e - 1)$

Answer

$0$

Answer

$2 \pi i e$

Question 40

Determine the value of the integral $\int_C \frac{1}{z}$ around the unit circle (positively oriented).

Accepted Answer

$2 \pi i$

Answer

$\pi i$

Answer

$2 \pi$

Answer

$0$

Question 41

Identify the singularities of the function $f(z) = \frac{1}{z(z - 1)}$.

Accepted Answer

$z = 0$ and $z = 1$

Answer

$z = 0$, $z = 1$ and $z = \infty$

Answer

$z = 0$ only

Answer

$z = 1$ only

Question 42

Which of the following integrals can be directly solved using Cauchy's integral formula?

Accepted Answer

$\int_C \frac{sin(z)}{(z - i)^2} dz$, where $C$ is the circle centered at the origin with radius $2$

Answer

$\int_C \frac{z^2}{z - 1} dz$, where $C$ is the circle centered at the origin with radius $1/2$

Answer

$\int_C e^{1/z} dz$, where $C$ is the circle centered at the origin with radius $1$

Question 43

For a function $f(z)$ with a simple pole at $z = a$, determine the formula for calculating the residue of $f(z)$ at $z = a$.

Accepted Answer

$lim_{z 	o a} (z-a)f(z)$

Answer

$\int_C f(z) dz$

Answer

$\frac{1}{2 \pi i} \int_C f(z) dz$, where $C$ is a small circle around $a$

Answer

$lim_{z 	o a} f(z)$

Question 44

Find the value of the improper integral $\int_{-\infty}^{\infty} \frac{1}{x^2 + 1} dx$ using complex integration techniques.

Accepted Answer

$\pi$

Answer

$2\pi$

Answer

$0$