IGNOU MSCMACS MMT 5 Complex Analysis Solved Assignment English

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MSCMACS 2026 - English

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MMT 5: Complex Analysis

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Title Name	IGNOU MSCMACS MMT 5 Complex Analysis Solved Assignment English
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
Assignment Code	MMT 5/Assignment-1/2026
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2026. Latest MMT 05 2026 Solved Assignment Solutions
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 ignouacademy-com-ignou-mmt-5-solved-assignment-html-p-english-91470

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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Course: M.Sc. Mathematics with Applications in Computer Science (MSCMACS)
Subject: MMT 5 - Complex Analysis
Session: January 2026 - December 2026
Language: English

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Product Information

Course Name	M.Sc. Mathematics with Applications in Computer Science
Course Code	MSCMACS
Subject Code	MMT 5
Subject	Complex Analysis
Programme	MASTER DEGREE PROGRAMMES Courses
Language	English
Material Type	Solved Assignment – Soft Copy / PDF

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