~~Rs.~~
Rs. 50

IGNOU MMT 5 2024 Solution

~~Rs.~~
Rs. 50

Last Date of Submission of IGNOU MMT-05 (MSCMACS) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Semester Wise
January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam).
July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).

Title Name	IGNOU MMT 5 Complex Analysis Solved Assignment 2024
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCMACS
Course Name	M.Sc. Mathematics with Applications in Computer Science
Subject Code	MMT 5
Subject Name	Complex Analysis
Year	2024
Session	-
Language	English Medium
Assignment Code	MMT-05/Assignmentt-1//2024
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2024. Latest MMT 05 2024 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MMT-05 (MSCMACS) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam). Semester Wise January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam). July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Assignment Code	MMT 5/2024

~~Rs.~~
Rs. 50

Questions Included in this Help Book

Ques 1.

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.

Ques 2.

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

Ques 3.

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.

Ques 4.

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

Ques 5.

Evaluate the following integrals:

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

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

Ques 6.

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 $r = 1$ ?

Ques 7.

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 $\lvert z \rvert \geq R$

Ques 8.

Find all solutions to the equation sin z = 5

Ques 9.

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.

Ques 10.

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

Ques 11.

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

Ques 12.

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

Ques 13.

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

Ques 14.

Find the radius of convergence of the following series

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

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

Ques 15.

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) $0 < \lvert z-3 \rvert < 2$

Ques 16.

Find the zeros and singularities of the function $f(z) = \frac{z}{4cos^2z - 1}$ in $\lvert z \rvert \leq 1$ . Also find the residue at the poles.

Ques 17.

Prove that the linear fractional transformation $\Phi (z) = \frac{2z-1}{2-z}$ maps the circle $c : \lvert z \rvert =1$ into itself. Also prove that $f(z)$ is conformal in $\overline{D} = \left \{ z : \lvert z \rvert \leq 1 \right \}$ .

Ques 18.

Find the image of the semi-infinite strip $x>0 , 0<y<1$ when $w= i/z$ . Sketch the strip and its image

Ques 19.

Show that there is only one linear fractional transformation that maps three given distinct points $z_1,z_2$ and $z_3$ in the extended $z$ plane onto three specified distinct points $w_1, w_2$ and $w_3$ in the extended $w$ plane.