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MMT 5: Complex Analysis

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 5/Assignment-1/2026
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2026. Latest MMT005 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
Format	Ready-to-Print PDF (.soft copy)

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MMT 5 2025 - English

Assignment

Course Code: MMT-005

Assignment Code:MMT-005/TMA/2025

Maximum Marks: 100

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

i) $ext{If}z=a+ib$ where a and b are integers, the $|1+z+z^2+cdots+z^n|geq|z|^nquad ext{if}a>0.$

ii) $ext{If}f(z) ext{and}overline{f(z)}$ 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 w(x,y) is harmonic in D.

iv) $lim_{n oinfty}(n!)^{1/n}=infty$

v) The inequality $|e^a-e^b|leq|a-b|$ holds for $a,bin D={w: ext{Re}wleq 0}.$

vii) If a power series $sum_{n=0}^{infty}a_n z^n$ converges for $|z|<1$ and if $b_{n}epsilon C$ is such that $|b_n|<n^2|a_n| ext{for all}ngeq 0, ext{then}sum_{n=0}^{infty}b_n z^n$ then converges for r | z | $<$ 1.

viii) If $f$ is entire and $f(z)=f(-z)$ for all, then there exists an entire function g such that $f(z)=g(z^2)quad ext{for all}zinmathbb{C}$

ix) A mobius transformation which maps the upper half plane ${z: ext{Im}z>0}$ onto itself and fixing 0, $infty$ and no other points, must be of the form $T_{z=}a_{z}$ for some $a>0$ and $alpha eq 1$

x) If $f$ is entire and Re $fleft(z ight)$ is bounded as $|z| oinfty$ then $f$ is constant.

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

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

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

3. a) Evaluate the following integrals:

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

ii) $ext{}quad I=int_0^{2pi}f(e^{i heta})sin^2( heta/2),d heta$

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

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

b) Find all solutions to the equation sin z=5.

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

b) Find all the singularities of the function $f(z)=expleft(frac{z}{sin z} ight)$

c) Evaluate $oint_Cfrac{dz}{z^2+1}$ de where e is the circle $|z|=4$

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

b) Evaluate $int_Cfrac{z^3+3}{z(z-i)^2},dz$ where e is the eight like figure shown in Fig. 1.

Image ignou-ignouacademy-com-ignou-mmt-5-solved-assignment-html-p-assignment-22561

c) Find the radius of convergence of the following series.

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

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

i) $0<|z-1|<2quad ext{and}$ ii) $0<|z-3|<2$

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

c) 22-1 Prove that the linear fractional transformation $phi(z)=frac{2z-1}{2-z}$ maps the circle $c:|z|=1$ into itself. Also prove that $fleft(z ight)$ is conformal in $overline{D}={z:|z|leq 1}.$

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

b) 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.

9. Evaluate the following integrals

a) $int_0^inftyfrac{x^2+2}{(x^2+1)(x^2+4)},dx$

b) $int_{-infty}^{infty}frac{sin^2 2x}{1+x^2},dx$

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