IGNOU MMT 5 SOLVED ASSIGNMENT

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IGNOU MMT 5 SOLVED ASSIGNMENT

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Rs. 123

IGNOU MMT 5 SOLVED ASSIGNMENT

~~Rs. 200~~
Rs. 123

Last Date of Submission of IGNOU MMT-05 (MSCMACS) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).
Semester Wise
January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam).
July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

Title Name	IGNOU MMT 5 SOLVED ASSIGNMENT
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	2026
Session
Language	English Medium
Assignment Code	MMT-05/Assignmentt-1//2026
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2026. Latest MMT 05 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MMT-05 (MSCMACS) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam). Semester Wise January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

~~Rs. 200~~
Rs. 123

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 $|1 + z + z^2 + \dots + z^n| \geq |z|^n$ if a > 0.
ii) If f(z) and $\overline{f(\bar{z})}$ 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 $|e^a - e^b| \leq |a - b|$ holds for $a, b \in D = \{w : \text{Re } w \leq 0\}$ .
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 $\sum_{n=0}^{\infty} a_n z^n$ converges for |z| < 1 and if $b_n \in \mathbb{C}$ is such that |b_n| < n² |a_n| for all $n \geq 0$ , then $\sum_{n=0}^{\infty} b_n z^n$ converges for |z| < 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 $z \in \mathbb{C}$ .
ix) A mobius transformation which maps the upper half plane $\{z : \text{Im } z > 0\}$ 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 $\text{Re } f(z)$ is bounded as $|z| \to \infty$ , then f is constant.

Ques 2.

a) If $f = u + iv$ is entire such that $u_x + v_y = 0$ in $\mathbb{C}$ then show that f has the form $f(z) = az + b$ where $a, \mathbf{b}$ are constants with $\operatorname{Re} a = 0$ .

Ques 3.

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

Ques 4.

Find the points where the function $f(z) = \frac{\operatorname{Log}(z + 4)}{z^2 + i}$ 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 $|z| = r (r \neq 1)$ under the mapping $w = f(z) = \frac{z - i}{z + i}$ . What happens when $r = 1$ ?

Ques 7.

If $p(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1} + z^n (n \geq 1)$ , then show that there exists a real R > 0 such that $2^{-1} |z|^n \leq |p(z)| \leq 2 |z|^n$ for $|z| \geq R$ .

Ques 8.

Find all solutions to the equation $\sin z = 5$ .

Ques 9.

Find the constant c such that $f(z) = \frac{1}{z^n + z^{n-1} + \dots + z^2 + z - n} + \frac{c}{z - 1}$ 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 $f(z) = \exp \left( \frac{z}{\sin z} \right)$

Ques 11.

Evaluate $\oint_c \frac{dz}{z^2 + 1}$ where c is the circle $|z| = 4$ .

Ques 12.

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

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

Ques 15.

Expand $f(z) = \frac{1}{(z-1)^2 (z-3)}$ in a Laurent series valid for

i) 0 < |z - 1| < 2 and $\quad \quad \quad \quad$ ii) 0 < |z - 3| < 2.

Ques 16.

Find the zeros and singularities of the function $f(z) = \frac{z}{4 \cos^2 z - 1}$ in $|z| \leq 1$ . 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 $c : |z| = 1$ into itself. Also prove that f(z) is conformal in $\bar{D} = \{z : |z| \leq 1\}$ .

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 z₁, z₂ and z₃ in the extended z plane onto three specified distinct points w₁, w₂ and w₃ in the extended w plane.

Ques 20.

Evaluate the following integrals

a) $\int_{0}^{\infty} \frac{x^2 + 2}{(x^2 + 1)(x^2 + 4)} \, dx$ .
b) $\int_{-\infty}^{\infty} \frac{\sin^2 2x}{1 + x^2} \, dx$ .

Ques 21.

Ques 22.

a) If $f = u + iv$ is entire such that $u_x + v_y = 0$ in $\mathbb{C}$ then show that f has the form $f(z) = az + b$ where $a, \mathbf{b}$ are constants with $\operatorname{Re} a = 0$ .

Ques 23.

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

Ques 24.

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

Ques 25.

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 26.

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

Ques 27.

If $p(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1} + z^n (n \geq 1)$ , then show that there exists a real R > 0 such that $2^{-1} |z|^n \leq |p(z)| \leq 2 |z|^n$ for $|z| \geq R$ .

Ques 28.

Find all solutions to the equation $\sin z = 5$ .

Ques 29.

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

Ques 30.

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

Ques 31.

Evaluate $\oint_c \frac{dz}{z^2 + 1}$ where c is the circle $|z| = 4$ .

Ques 32.

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

Ques 33.

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

Ques 34.

Find the radius of convergence of the following series.
i) $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k!} (z - 1 - i)^k$ $\quad \quad$ ii) $\sum_{k=1}^{\infty}\left(\frac{6k+1}{2k+5}\right)^k(z-2i)^k$

Ques 35.

Expand $f(z) = \frac{1}{(z-1)^2 (z-3)}$ in a Laurent series valid for

i) 0 < |z - 1| < 2 and $\quad \quad \quad \quad$ ii) 0 < |z - 3| < 2.

Ques 36.

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

Ques 37.

Prove that the linear fractional transformation $\phi(z) = \frac{2z - 1}{2 - z}$ maps the circle $c : |z| = 1$ into itself. Also prove that f(z) is conformal in $\bar{D} = \{z : |z| \leq 1\}$ .

Ques 38.

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

Ques 39.

Ques 40.

Evaluate the following integrals

a) $\int_{0}^{\infty} \frac{x^2 + 2}{(x^2 + 1)(x^2 + 4)} \, dx$ .
b) $\int_{-\infty}^{\infty} \frac{\sin^2 2x}{1 + x^2} \, dx$ .

~~Rs. 200~~
Rs. 123

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Course Name	M.Sc. Mathematics with Applications in Computer Science
Course Code	MSCMACS
Programm	MASTER DEGREE PROGRAMMES Courses
Language	English

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