Solve your IGNOU Doubts

Question:

Expand $f(z) = \frac{1}{(z-1)^2(z-3))}$ in a Laurent series valid for i) $0 < \lvert z-1 \rvert < 2$ and ii) $0 < \lvert z-3 \rvert < 2$

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Question:

Find the radius of convergence of the following series

i) $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k!} (z-1-i)^k$

ii) $\sum_{k=1}^{\infty}\left ( \frac{6k + 1}{2k + 5} \right )^k\left ( z - 2i \right )^k$

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Question:

Evaluate $\int_{C}^{} \frac{z^3 + 3}{z(z-i)^2}dz$ , where $c$ is the eight like figure shown in Fig. 1

Image ignouassignments-ignouacademy-com--p-your-86324

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Question:

Find the maximum modulus of $f(z) = 2z + 5i$ on the closed circular region defined by $\lvert z \rvert \leq 2$

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Question:

Evaluate $\oint_{C}^{}\frac{dz}{z^2 + 1}$ where $c$ is the circle $\lvert z \rvert = 4$ .

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Question:

Find all the singularities of the function $f(z) = exp(\frac{z}{sin z})$

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Question:

Find the constant c such that $f(z) = \frac{1}{z^n + z^{n-1} + .... + z^2 + z^{-n}} + \frac{C}{z-1}$ can be extended to be analytic at $z =1$ , when $n \in \mathbb{N}$ is fixed.

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Question:

Find all solutions to the equation sin z = 5

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Question:

If $p(z) = a_0 + a_1z + ..... + a_{n-1}z^{n-1} + z^n(n \geq 1)$ , then show that there exists a real $R > 0$ such that $2^{-1}\lvert z \rvert^n \leq \lvert p(z) \rvert \leq 2 \lvert z \rvert^n$ for $\lvert z \rvert \geq R$

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Question:

Find the image of the circle $\lvert z \rvert = r (r \neq 1)$ under the mapping $w = f(z) = \frac{z-i}{z+i}$ . What happens when $r = 1$ ?

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Question:

Evaluate the following integrals:

i) $I = \int_{0}^{2\pi}f(e^{i\theta}) cos^2(\theta/2) d\theta$

ii) $I = \int_{0}^{2\pi}f(e^{i\theta}) sin^2(\theta/2) d\theta$

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Question:

Find the points where the function $f(z) = \frac{\log (z+4)}{z^2 + i}$ is not analytic.

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Question:

Consider $f(z) = z^2 - z$ and the closed circular region $R = \left \{ z:\lvert z \rvert \leq 1 \right \}$ . Find points in $R$ where $\lvert f(z) \rvert$ has its maximum and minimum values.

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Question:

If f = u + iv is entire such that ux + vy = 0 in C then show that f has the form f (z) = az + b where a, b are constants with Re a = 0

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Question:

Determine whether each of the following statement is true or false. Justify your answer with a short proof or a counter example

i). if $z = a + ib$ , where a and b are integers, then $\lvert 1 + z + z^2 + .... + z^n \rvert \geq \lvert z \rvert ^n$ if a>0

ii) If $f(z)$ and $\overline{f(z)}$ are analytic functions in $a$ domain, then
$f$ is necessarily a constant.
iii) A real-valued function u(x, y) is harmonic in $D$ iff u(x, − y)
is harmonic in $D$ .
iv) $\lim_{n \to \infty }(n!)^{1/n} = \infty$
v) The inequality $\lvert e^a -e^b \rvert \leq \lvert a - b\rvert$ holds for $a,b \in D = \left \{ w : Re\: w \leq 0 \right \}$ .
vi) If $f(z) = \sum_{n=0}^{\infty}a_n(z-a)^n$ has the property that $\sum_{n=0}^{\infty}f^{(n)}(a)$ converges, then $f$ is necessarily an entire function.
vii) If a power series $\sum_{n=0}^{\infty}a_nz^n$ converges for $\lvert z \rvert< 1$ and if
$b_n \in C$ is such that $\lvert b_n \rvert < n^2 \lvert a_n \rvert$ for all $n \geq 0$ , then $\sum_{n=0}^{\infty}b_nz^n$ converges for $\lvert z \rvert < 1$ .
viii) If $f$ is entire and $f (z) = f (-z)$ for all z, then there exists an entire function g such that $f (z) = g(z^2)$ for all $z \in C$ .
ix) A mobius transformation which maps the upper half plane $\left \{ z : Im \: z > 0 \right \}$ onto itself and fixing $0, \infty$ and no other points, must be of the form $Tz = \alpha z$ for some $\alpha > 0$ and $\alpha \neq 1$ .
x) If $f$ is entire and Re f(z) is bounded as $\lvert z \rvert \to \infty$ , then
$f$ is constant.