Solve your IGNOU Doubts

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.

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Question:

Find the SVD of the following matrices:

i). $\begin{bmatrix} -1 & 1 & 1\\ 1 & 1 & 0\\ \end{bmatrix}$

ii). $\begin{bmatrix} -1 & 1 \\ 1 & 1 \\ 1 & 2 \end{bmatrix}$

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Question:

Find the QR decomposition of the matrix

$\begin{bmatrix} 2 & -2 & 1\\ 2 & 2 & 1\\ 0& 1& 1\\ 1 & 0 & 1 \end{bmatrix}$

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Question:

Check which of the following matrices is positive definite and which is positive semi-definite:

$A = \begin{bmatrix} 1 & 1 & 0\\ 1 & 2 & 1\\ 0 & 1 & 1 \end{bmatrix}, B \begin{bmatrix} 2 & 0 & 1\\ 0 & 2 & -1\\ 1 & -1 & 3 \end{bmatrix}$

Also, find the square root of the positive definite matrix.

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Question:

Use least squares method to find a quadratic polynomial that fits the following data: (-2, 15.7), (-1, 6.7), (0, 2.7), (1, 3.7), (2, 9.7).

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Question:

Let

$A = \begin{bmatrix} 2 & 2 & 1\\ -1 & -1 & 2\\ 0 & 0 & -2 \end{bmatrix}$

Find a unitary matrix $U$ such that $U^*AU$ is upper triangular.

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Question:

Solve the following system of differential equations:

$\frac{dy(t)}{dt} = Ay(t)$ with $y(0) = \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}$ , Where $A = \begin{bmatrix} 2 &-5 &-11 \\ 0 &-2 &-9 \\ 0 &1 &4 \end{bmatrix}$

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Question:

Let M and T be a metro city and a nearby district town, respectively. Our government is trying to develop infrastructure in T so that people shift to T. Each year 15% of T’s population moves to M and 10% of M’s population moves to T. What is the long term effect of on the population of M and T? Are they likely to stabilise?

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Question:

Find the Jordan canonical form J for

$B = \begin{bmatrix} -1 &0 & -2 & -4 \\ 2 & 1 & 2 & 4 \\ -4 & 2 & -1 & -4\\ 2 & -1 & 1 & 3 \end{bmatrix}$

Also, find a matrix P such that J = P^-1BP

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Question:

Can A be similar to A + I? Give reasons for your answer

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Question:

If C and D are $n \times n$ matrices such that CD = -DC and D^-1 exists, then show that C is similar to -D. Hence show that the eigenvalues of C must come in plus-minus pairs.

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Question:

Let $T : C^2 \to C^2 : T\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x + 2y - iz \\ 2y + iz \\ ix + z - 2z \end{bmatrix}$ . FInd [T]_B , [T]_B' and P where

$B = \left \{ \begin{bmatrix} 0\\ i\\ 0 \end{bmatrix}, \begin{bmatrix} i\\ 1\\ -1 \end{bmatrix} , \begin{bmatrix} 0\\ 0\\ 2 \end{bmatrix} \right \}$ , $B' = \left \{ \begin{bmatrix} 1\\ -i\\ 1 \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix} , \begin{bmatrix} 1\\ i\\ 0 \end{bmatrix} \right \}$ ,