which one of the following is a maximum principle for harmonic functions?

A harmonic function cannot attain a maximum value in the interior of its domain.

A harmonic function can attain a maximum value at any point in its domain.

Harmonic functions are always increasing in their domain.

A harmonic function can have multiple maximum values in the interior of its domain.

what is the purpose of Poisson's formula in complex analysis?

To relate the values of harmonic functions inside a domain to their values on the boundary of that domain.

To calculate the roots of polynomials.

To determine the convergence of series.

A region $$\Omega$$ is said to be symmetric if

A region $$\Omega$$ is symmetric if it is invariant under transformations like reflection or rotation.

A region $$\Omega$$ is symmetric if it is colored uniformly.

A region $$\Omega$$ is symmetric if it contains only straight lines.

A region $$\Omega$$ is symmetric if it has a constant area.

Suppose that the function $$f_n\left(z\right)$$ are analytic and not equal to zero in a region $$\Omega$$ , and $$f_n\left(z\right)$$ converges to $$f\left(z\right)$$ uniformly on every compact subset of $$\Omega$$ . Then choose the correct statement given below

f(z) is not equal to zero in $$\Omega$$ .

f(z) converges to zero in $$\Omega$$

f(z) is not analytic in $$\Omega$$

Complex Analysis (MP23CC1) - Quiz II

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MULTIPLE CHOICE QUESTION

3 mins • 1 pt

If $f\left(z\right)$ is analytic except for isolated singularities $a_j$ in a region $\Omega$ , then $\frac{1}{2\pi i}\int_{\gamma}^{ }f\left(z\right)dz$ is equal to

$Res_{z=a_j\ \ ^{f\left(z\right)}}$

$\sum_j^{ }\ \ Res_{z=a_j\ \ ^{f\left(z\right)}}$

$\sum_j^{ }\ n\left(\gamma,\ a_j\right)\ Res_{z=a_j\ \ ^{f\left(z\right)}}$

None of these

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

The value of $\int_0^{\pi}\ \log\ \sin x\ dx$ is

$\pi\ \log\ 2$

$-\pi\ \log\ 2$

$\log\ 2$

$-\log\ 2$

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

The pole of the function $\frac{1}{\sin\ z}$

o and $\pi$

$-n\pi$ , where n is an integer

$n\pi$ , where n is an integer

MULTIPLE CHOICE QUESTION

5 mins • 2 pts

The residue of the function $\frac{1}{\left(z+2\right)\left(z+3\right)}$

$\frac{4\pi i}{5}$

$\frac{2\pi i}{5}$

$4\pi i$

$2\pi i$

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

If $z=a$ is a pole of $f\left(z\right),$ then residue of $f\left(z\right)$ at $z=a$ is

The residue of f(z) at z=a is given by f(a).

The residue of f(z) at z=a is always zero.

The residue of f(z) at z=a is calculated using the integral of f(z) around a.

The residue of f(z) at z=a is given by Res(f, a) = lim (z -> a) (z - a) f(z).

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

what does the argument principle shows that

The argument principle shows the relationship between the number of zeros and poles of a function within a contour.

It provides a method for calculating integrals along a contour.

It shows the continuity of a function over a contour.

It demonstrates the symmetry of poles and zeros in complex functions.

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

what is the meaning behind the statement of Rouche's Theorem

Rouche's Theorem states that all complex functions have at least one zero.

Rouche's Theorem is used to find the maximum value of a function on a contour.

Rouche's Theorem applies only to real-valued functions and their derivatives.

Rouche's Theorem provides a method to count the zeros of complex functions within a contour by comparing them to another function.

MULTIPLE SELECT QUESTION

5 mins • 2 pts

The value of * $du$ is

$\frac{du}{dx\ }dy-\frac{du}{dy}dx$

$-\frac{du}{dy}dx+\frac{du}{dx\ }dy$

$\frac{du}{dx\ }dy+\frac{du}{dy}dx$

$-\frac{du}{dx\ }dy+\frac{du}{dy}dx$

10.

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

If $C$ is a circle with radius $r$ , then the value of * $du$ is

$r\ d\theta$

$\frac{du}{dr}r\ d\theta$

$\frac{du}{d\theta}r\ dr$

$\frac{1}{r}d\theta$

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Complex Analysis (MP23CC1) - Quiz II

If f(z)f\left(z\right)f(z) is analytic except for isolated singularities aja_jaj​ in a region Ω\OmegaΩ , then 12πi∫γf(z)dz\frac{1}{2\pi i}\int_{\gamma}^{ }f\left(z\right)dz2πi1​∫γ​f(z)dz is equal to

The value of ∫0π log⁡ sin⁡x dx\int_0^{\pi}\ \log\ \sin x\ dx∫0π​ log sinx dx is

The pole of the function 1sin⁡ z\frac{1}{\sin\ z}sin z1​

The residue of the function 1(z+2)(z+3)\frac{1}{\left(z+2\right)\left(z+3\right)}(z+2)(z+3)1​

If z=az=az=a is a pole of f(z),f\left(z\right),f(z), then residue of f(z)f\left(z\right)f(z) at z=az=az=a is

what does the argument principle shows that

what is the meaning behind the statement of Rouche's Theorem

Among the following, which one is the simplest harmonic function?

The value of * dududu is

If CCC is a circle with radius rrr , then the value of * dududu is

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Discover more resources for Mathematics

If $f\left(z\right)$ is analytic except for isolated singularities $a_j$ in a region $\Omega$ , then $\frac{1}{2\pi i}\int_{\gamma}^{ }f\left(z\right)dz$ is equal to

The value of $\int_0^{\pi}\ \log\ \sin x\ dx$ is

The pole of the function $\frac{1}{\sin\ z}$

The residue of the function $\frac{1}{\left(z+2\right)\left(z+3\right)}$

If $z=a$ is a pole of $f\left(z\right),$ then residue of $f\left(z\right)$ at $z=a$ is

The value of * $du$ is

If $C$ is a circle with radius $r$ , then the value of * $du$ is