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

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

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

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

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

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

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

Boundary of closed and bounded set E

Interior of closed and bounded set E

Choose the correct statement(s)

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

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 $$\gamma$$

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

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

Which of the following statement is/ are false?

A non constant function maps open onto open sets

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

Any analytic and bounded function must reduce to a constant

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

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

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

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

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

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

What is meant by zero sum in a chain?

sum of all coefficients equal to zero

at least one of the coefficient equal to zero

sum of all coefficients equal to zero

Choose the correct statement(s)

sum of closed curves is a cycle

a region without hole is simply connected

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

Generalization of Cauchy's Theorem

Cauchy's Theorem for Closed Curves

Complex Analysis (MP233CC1) - Quiz 1

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Mathematics

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MULTIPLE SELECT QUESTION

5 mins • 1 pt

Which of the following is/are correct?

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

The curves will have different lengths.

The curves will have different shapes.

The net area enclosed by both curves will be infinite.

The area enclosed by both curves will be zero.

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

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

True

False

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

True

False

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

4πi

πi

3πi

2πi

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

The winding number of a point inside the circle is

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

Representation Formula

Cauchy Integral Formula

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Complex Analysis (MP233CC1) - Quiz 1

Which of the following is/are correct?

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

The winding number of a point inside the circle is

The value of Cauchy's Estimate is

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