Quiz about Complex Analysis (MP233CC1) - Quiz 1

Question 1

Which of the following is/are correct?

Accepted Answer

If $$f\left(t\right)=u\left(t\right)+iv\left(t\right)$$ is a continuous function defined in an interval $$\left(a,\ b\right),\ $$ then the value of the integral $$f\left(t\right)$$ is defined to be $$\int_a^bf\left(t\right)dt\ =\int_a^bu\left(t\right)dt+i\int_a^bv\left(t\right)dt$$

Accepted Answer

If $$f\left(t\right)=u\left(t\right)+iv\left(t\right)$$ is a continuous function defined in an interval $$\left(a,\ b\right),\ $$ then the value of the integral $$f\left(t\right)$$ is defined to be $$\int_a^bf\left(t\right)dt\ =\int_a^b\left(u\left(t\right)dt+iv\left(t\right)\right)dt$$

Accepted Answer

If $$\gamma$$ is a piecewise differentiable arc with $$z=z\left(t\right),\ a\le t\le b$$ and $$f\left(z\right)$$ is continuous on $$\gamma,$$ then $$\int_{\gamma}^{ }f\left(z\right)dz=\int_a^bf\left(z\left(t\right)\right)z$$

Answer

If $$\gamma$$ is a piecewise differentiable arc with $$z=z\left(t\right),\ a\le t\le b$$ and $$f\left(z\right)$$ is continuous on $$\gamma,$$ then $$\int_{\gamma}^{ }f\left(z\right)dz=\int_a^bf\left(u\left(t\right)\right)u$$

Question 2

If the closed curves $$\gamma$$ and $$-\gamma$$ have the same end points, then

Accepted Answer

The area enclosed by both curves will be zero.

Answer

The curves will have different lengths.

Answer

The curves will have different shapes.

Answer

The net area enclosed by both curves will be infinite.

Question 3

True or False: The integral value of any analytic function in the rectangle R is zero.

Accepted Answer

False

Answer

True

Question 4

True or False: The integral value of an analytic function $$f\left(z\right)$$ in an open disc $$\Delta$$ , over the curve $$\gamma$$ in $$\Delta$$ is zero.

Accepted Answer

False

Answer

True

Question 5

The winding number of a point inside the circle is

Accepted Answer

1

Answer

2

Answer

3

Answer

0

Question 6

$$f(a)=\int_{\gamma\ \ \ \ \ }^{ }\ \frac{f\left(z\right)}{z-a}dz$$ is known as

Accepted Answer

Representation Formula

Answer

Winding number of a with respect to $$\gamma$$

Answer

Representation of $$\gamma$$

Answer

Cauchy Integral Formula

Question 7

The value of Cauchy's Estimate is

Accepted Answer

$$\frac{n!M}{r^n}$$

Answer

$$\frac{M}{r}$$

Answer

$$\frac{M}{r^n}$$

Answer

$$\frac{n!M}{r}$$

Question 8

True or False: Any function $$f\left(z\right)$$ can be expressed as a power series about the point $$a$$

Accepted Answer

False

Answer

True

Question 9

True or False: Let $$x,\ y$$ be zeros of an analytic function. Then there exist an neighborhood which contains both $$x$$ and $$y$$ .

Accepted Answer

False

Answer

True

Question 10

Find the pole(s) for the function $$f\left(z\right)=\frac{z^2}{\left(z+1\right)\left(z-2\right)}$$

Accepted Answer

The poles are z = -1 and z = 2.

Answer

The poles are z = 1 and z = 2.

Answer

The poles are z = 1 and z = -2.

Answer

The poles are z = -1 and z = -2.

Question 11

If $$f\left(z\right)$$ is analytic and non-constant in a region $$\Omega$$ , then which of the value of $$f\left(z\right)$$ has no maximum in $$\Omega$$ ?

Accepted Answer

$$\left|f\left(z\right)\right|$$

Answer

$$f\left(z\right)$$

Answer

$$\left(f\left(z\right)\right)^2$$

Answer

$$\left(f\left(z\right)\right)^{\frac{1}{2}}$$

Question 12

In which place, the maximum value of modulus of $$f\left(z\right)$$ attains?

Accepted Answer

Boundary of closed and bounded set E

Answer

Interior of closed and bounded set E

Answer

Interior of closed set E

Answer

Boundary of closed set E

Question 13

Choose the correct statement(s)

Accepted Answer

The number of zeros enclosed by $$\gamma$$ is the winding number of 0 with respect to image of $$\gamma$$

Accepted Answer

The number of zeros of $$f\left(z-a\right)$$ enclosed by $$\gamma$$ is the winding number of a with respect to the image of $$\gamma$$

Answer

The number of zeros enclosed by $$\gamma$$ is the winding number of 0 with respect to $$\gamma$$

Answer

The number of zeros of $$f\left(z-a\right)$$ enclosed by $$\gamma$$ is the winding number of a with respect to $$\gamma$$

Question 14

Which of the following statement is/ are false?

Accepted Answer

A non constant function maps open onto open sets

Accepted Answer

If $$f\left(z\right)$$ is analytic at $$z$$ , then it maps a neighbourhood of $$z$$ conformally and topologically onto a region

Answer

Any analytic and bounded function must reduce to a constant

Answer

$$f_n\left(z\right)=\frac{1}{2\pi i}\int_C^{ }\ \ \frac{f\left(\xi\right)}{\left(\xi-a\right)^n\ \left(\xi-z\right)}d\xi$$ is the representation for $$f_n\left(z\right)$$

Question 15

Choose the characterization for a region $$\Omega$$ to be simply connected

Accepted Answer

Winding number $$n\left(\gamma,\ a\right)=0$$ for all cycles $$\gamma$$ in $$\Omega$$ and for all the point $$a$$ does not belong to $$\Omega$$

Answer

Winding number $$n\left(\gamma,\ a\right)=0$$ for all the point $$a$$ belong to $$\Omega$$

Answer

Winding number $$n\left(\gamma,\ a\right)=0$$ for all cycles $$\gamma$$ in $$\Omega$$ and for all the point $$a$$ belong to $$\Omega$$

Answer

Winding number $$n\left(\gamma,\ a\right)=0$$ for all cycles $$\gamma$$ in $$\Omega$$

Question 16

Which implies which: Chain, Cylce

Accepted Answer

cycle implies chain

Answer

Chain implies cycle

Question 17

What is meant by zero sum in a chain?

Accepted Answer

sum of all coefficients equal to zero

Answer

at least one of the coefficient equal to zero

Answer

all coefficient must be zero

Answer

sum of all coefficients equal to zero

Question 18

Choose the correct statement(s)

Accepted Answer

sum of arcs is chain

Accepted Answer

sum of closed curves is a cycle

Accepted Answer

a region without hole is simply connected

Accepted Answer

half plane is simply connected

Question 19

If $$f\left(z\right)$$ is analytic in $$\Omega$$ , then $$\int_{\Upsilon}^{ }\ f\left(z\right)dz=0$$ for every cycle $$\gamma$$ which is homologous to zero in $$\Omega$$ . This statement is known as

Accepted Answer

Generalization of Cauchy's Theorem

Answer

Cauchy's Integral Formula

Answer

Cauchy's Residue Theorem

Answer

Cauchy's Theorem for Closed Curves

Complex Analysis (MP233CC1) - Quiz 1

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