IGNOU MMT-005 Solved Assignment 2024 for M.Sc. MACS

IGNOU MMT-005 Solved Assignment 2024 | M.Sc. MACS

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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IGNOU MMT-05 Assignment Question Paper 2024

mmt-005-solved-assignment-2024-qp-b5864dd3-9e6a-403d-817d-3f61b693b950

mmt-005-solved-assignment-2024-qp-b5864dd3-9e6a-403d-817d-3f61b693b950

  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 + i b z = a + i b z=a+ibz=a+i bz=a+ib, where a a aaa and b b bbb are integers, then | 1 + z + z 2 + + z n | | z | n 1 + z + z 2 + + z n | z | n |1+z+z^(2)+cdots+z^(n)| >= |z|^(n)\left|1+z+z^2+\cdots+z^n\right| \geq|z|^n|1+z+z2++zn||z|n if a > 0 a > 0 a > 0a>0a>0.
    ii) If f ( z ) f ( z ) f(z)f(z)f(z) and f ( z ) ¯ f ( z ) ¯ bar(f(z))\overline{f(z)}f(z)¯ are analytic functions in a a aaa domain, then f f fff is necessarily a constant.
    iii) A real-valued function u ( x , y ) u ( x , y ) u(x,y)u(x, y)u(x,y) is harmonic in D D DDD iff u ( x , y ) u ( x , y ) u(x,-y)u(x,-y)u(x,y) is harmonic in D D DDD.
    iv) lim n ( n ! ) 1 / n = lim n ( n ! ) 1 / n = lim_(n rarr oo)(n!)^(1//n)=oo\lim _{n \rightarrow \infty}(n !)^{1 / n}=\inftylimn(n!)1/n=.
    v) The inequality | e a e b | | a b | e a e b | a b | |e^(a)-e^(b)| <= |a-b|\left|e^a-e^b\right| \leq|a-b||eaeb||ab| holds for a , b D = { w : Re w 0 } a , b D = { w : Re w 0 } a,b in D={w:Re w <= 0}a, b \in D=\{w: \operatorname{Re} w \leq 0\}a,bD={w:Rew0}.
    vi) If f ( z ) = n = 0 a n ( z a ) n f ( z ) = n = 0 a n ( z a ) n f(z)=sum_(n=0)^(oo)a_(n)(z-a)^(n)f(z)=\sum_{n=0}^{\infty} a_n(z-a)^nf(z)=n=0an(za)n has the property that n = 0 f ( n ) ( a ) n = 0 f ( n ) ( a ) sum_(n=0)^(oo)f^((n))(a)\sum_{n=0}^{\infty} f^{(n)}(a)n=0f(n)(a) converges, then f f fff is necessarily an entire function.
    vii) If a power series n = 0 a n z n n = 0 a n z n sum_(n=0)^(oo)a_(n)z^(n)\sum_{n=0}^{\infty} a_n z^nn=0anzn converges for | z | < 1 | z | < 1 |z| < 1|z|<1|z|<1 and if b n C b n C b_(n)inCb_n \in \mathbb{C}bnC is such that | b n | < n 2 | a n | b n < n 2 a n |b_(n)| < n^(2)|a_(n)|\left|b_n\right|<n^2\left|a_n\right||bn|<n2|an| for all n 0 n 0 n >= 0n \geq 0n0, then n = 0 b n z n n = 0 b n z n sum_(n=0)^(oo)b_(n)z^(n)\sum_{n=0}^{\infty} b_n z^nn=0bnzn converges for | z | < 1 | z | < 1 |z| < 1|z|<1|z|<1.
    viii) If f f fff is entire and f ( z ) = f ( z ) f ( z ) = f ( z ) f(z)=f(-z)f(z)=f(-z)f(z)=f(z) for all z z zzz, then there exists an entire function g g ggg such that f ( z ) = g ( z 2 ) f ( z ) = g z 2 f(z)=g(z^(2))f(z)=g\left(z^2\right)f(z)=g(z2) for all z C z C z inCz \in \mathbb{C}zC.
    ix) A mobius transformation which maps the upper half plane { z : Im z > 0 } { z : Im z > 0 } {z:Im z > 0}\{z: \operatorname{Im} z>0\}{z:Imz>0} onto itself and fixing 0 , 0 , 0,oo0, \infty0, and no other points, must be of the form T z = α z T z = α z Tz=alpha zT z=\alpha zTz=αz for some α > 0 α > 0 alpha > 0\alpha>0α>0 and α 1 α 1 alpha!=1\alpha \neq 1α1.
    x) If f f fff is entire and Re f ( z ) Re f ( z ) Re f(z)\operatorname{Re} f(z)Ref(z) is bounded as | z | | z | |z|rarr oo|z| \rightarrow \infty|z|, then f f fff is constant.
  2. a) If f = u + i v f = u + i v f=u+ivf=u+i vf=u+iv is entire such that u x + v y = 0 u x + v y = 0 u_(x)+v_(y)=0u_x+v_y=0ux+vy=0 in C C C\mathbb{C}C then show that f f fff has the form f ( z ) = a z + b f ( z ) = a z + b f(z)=az+bf(z)=a z+bf(z)=az+b where a , b a , b a,ba, \mathbf{b}a,b are constants with Re a = 0 Re a = 0 Re a=0\operatorname{Re} a=0Rea=0.
    b) Consider f ( z ) = z 2 z f ( z ) = z 2 z f(z)=z^(2)-zf(z)=z^2-zf(z)=z2z and the closed circular region R = { z : | z | 1 } R = { z : | z | 1 } R={z:|z| <= 1}R=\{z:|z| \leq 1\}R={z:|z|1}. Find points in R R RRR where | f ( z ) | | f ( z ) | |f(z)||f(z)||f(z)| has its maximum and minimum values.
    c) Find the points where the function f ( z ) = log ( z + 4 ) z 2 + i f ( z ) = log ( z + 4 ) z 2 + i f(z)=(log(z+4))/(z^(2)+i)f(z)=\frac{\log (z+4)}{z^2+i}f(z)=log(z+4)z2+i is not analytic.
  3. a) Evaluate the following integrals:
    i) I = 0 2 π f ( e i θ ) cos 2 ( θ / 2 ) d θ I = 0 2 π f e i θ cos 2 ( θ / 2 ) d θ I=int_(0)^(2pi)f(e^(i theta))cos^(2)(theta//2)d thetaI=\int_0^{2 \pi} f\left(e^{i \theta}\right) \cos ^2(\theta / 2) d \thetaI=02πf(eiθ)cos2(θ/2)dθ.
    ii) I = 0 2 π f ( e i θ ) sin 2 θ / 2 d θ I = 0 2 π f e i θ sin 2 θ / 2 d θ quad I=int_(0)^(2pi)f(e^(i theta))sin^(2)theta//2d theta\quad I=\int_0^{2 \pi} f\left(e^{i \theta}\right) \sin ^2 \theta / 2 d \thetaI=02πf(eiθ)sin2θ/2dθ.
    b) Find the image of the circle | z | = r ( r 1 ) | z | = r ( r 1 ) |z|=r(r!=1)|z|=r(r \neq 1)|z|=r(r1) under the mapping w = f ( z ) = z i z + i w = f ( z ) = z i z + i w=f(z)=(z-i)/(z+i)w=f(z)=\frac{z-i}{z+i}w=f(z)=ziz+i. What happens when r = 1 r = 1 r=1r=1r=1 ?
  4. a) If p ( z ) = a 0 + a 1 z + + a n 1 z n 1 + z n ( n 1 ) p ( z ) = a 0 + a 1 z + + a n 1 z n 1 + z n ( n 1 ) p(z)=a_(0)+a_(1)z+cdots+a_(n-1)z^(n-1)+z^(n)(n >= 1)p(z)=a_0+a_1 z+\cdots+a_{n-1} z^{n-1}+z^n(n \geq 1)p(z)=a0+a1z++an1zn1+zn(n1), then show that there exists a real R > 0 R > 0 R > 0R>0R>0 such that 2 1 | z | n | p ( z ) | 2 | z | n 2 1 | z | n | p ( z ) | 2 | z | n 2^(-1)|z|^(n) <= |p(z)| <= 2|z|^(n)2^{-1}|z|^n \leq|p(z)| \leq 2|z|^n21|z|n|p(z)|2|z|n for | z | R | z | R |z| >= R|z| \geq R|z|R.
    b) Find all solutions to the equation sin z = 5 sin z = 5 sin z=5\sin z=5sinz=5.
  5. a) Find the constant c c ccc such that f ( z ) = 1 z n + z n 1 + + z 2 + z n + c z 1 f ( z ) = 1 z n + z n 1 + + z 2 + z n + c z 1 f(z)=(1)/(z^(n)+z^(n-1)+cdots+z^(2)+z^(-n))+(c)/(z-1)f(z)=\frac{1}{z^n+z^{n-1}+\cdots+z^2+z^{-n}}+\frac{c}{z-1}f(z)=1zn+zn1++z2+zn+cz1 can be extended to be analytic at z = 1 z = 1 z=1z=1z=1, when n N n N n inNn \in \mathbb{N}nN is fixed.
    b) Find all the singularities of the function f ( z ) = exp ( z sin z ) f ( z ) = exp z sin z f(z)=exp((z)/(sin z))f(z)=\exp \left(\frac{z}{\sin z}\right)f(z)=exp(zsinz).
    c) Evaluate C d z z 2 + 1 C d z z 2 + 1 oint_(C)(dz)/(z^(2)+1)\oint_C \frac{d z}{z^2+1}Cdzz2+1 where c c ccc is the circle | z | = 4 | z | = 4 |z|=4|z|=4|z|=4.
  6. a) Find the maximum modulus of f ( z ) = 2 z + 5 i f ( z ) = 2 z + 5 i f(z)=2z+5if(z)=2 z+5 if(z)=2z+5i on the closed circular region defined by | z | 2 | z | 2 |z| <= 2|z| \leq 2|z|2.
    b) Evaluate C z 3 + 3 z ( z i ) 2 d z C z 3 + 3 z ( z i ) 2 d z int _(C)(z^(3)+3)/(z(z-i)^(2))dz\int_C \frac{z^3+3}{z(z-i)^2} d zCz3+3z(zi)2dz, where c c ccc is the eight like figure shown in Fig. 1.
    original image

    c) Find the radius of convergence of the following series.
    i) k = 1 ( 1 ) k + 1 k ! ( z 1 i ) k k = 1 ( 1 ) k + 1 k ! ( z 1 i ) k quadsum_(k=1)^(oo)((-1)^(k+1))/(k!)(z-1-i)^(k)\quad \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k !}(z-1-i)^kk=1(1)k+1k!(z1i)k
    ii) k = 1 ( 6 k + 1 2 k + 5 ) k ( z 2 i ) k k = 1 6 k + 1 2 k + 5 k ( z 2 i ) k quadsum_(k=1)^(oo)((6k+1)/(2k+5))^(k)(z-2i)^(k)\quad \sum_{k=1}^{\infty}\left(\frac{6 k+1}{2 k+5}\right)^k(z-2 i)^kk=1(6k+12k+5)k(z2i)k
  7. a) Expand f ( z ) = 1 ( z 1 ) 2 ( z 3 ) f ( z ) = 1 ( z 1 ) 2 ( z 3 ) f(z)=(1)/((z-1)^(2)(z-3))f(z)=\frac{1}{(z-1)^2(z-3)}f(z)=1(z1)2(z3) in a Laurent series valid for
    i) 0 < | z 1 | < 2 0 < | z 1 | < 2 0 < |z-1| < 20<|z-1|<20<|z1|<2 and
    ii) 0 < | z 3 | < 2 0 < | z 3 | < 2 quad0 < |z-3| < 2\quad 0<|z-3|<20<|z3|<2.
    b) Find the zeros and singularities of the function f ( z ) = z 4 cos 2 z 1 f ( z ) = z 4 cos 2 z 1 f(z)=(z)/(4cos^(2)z-1)f(z)=\frac{z}{4 \cos ^2 z-1}f(z)=z4cos2z1 in | z | 1 | z | 1 |z| <= 1|z| \leq 1|z|1. Also find the residue at the poles.
    c) Prove that the linear fractional transformation ϕ ( z ) = 2 z 1 2 z ϕ ( z ) = 2 z 1 2 z phi(z)=(2z-1)/(2-z)\phi(z)=\frac{2 z-1}{2-z}ϕ(z)=2z12z maps the circle c : | z | = 1 c : | z | = 1 c:|z|=1c:|z|=1c:|z|=1 into itself. Also prove that f ( z ) f ( z ) f(z)f(z)f(z) is conformal in D ¯ = { z : | z | 1 } D ¯ = { z : | z | 1 } bar(D)={z:|z| <= 1}\bar{D}=\{z:|z| \leq 1\}D¯={z:|z|1}.
  8. a) Find the image of the semi-infinite strip x > 0 , 0 < y < 1 x > 0 , 0 < y < 1 x > 0,0 < y < 1x>0,0<y<1x>0,0<y<1 when w = i / z w = i / z w=i//zw=i / zw=i/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 z 1 , z 2 z_(1),z_(2)z_1, z_2z1,z2 and z 3 z 3 z_(3)z_3z3 in the extended z z zzz plane onto three specified distinct points w 1 , w 2 w 1 , w 2 w_(1),w_(2)w_1, w_2w1,w2 and w 3 w 3 w_(3)w_3w3 in the extended w w www plane.
  9. Evaluate the following integrals
    a) 0 x 2 + 2 ( x 2 + 1 ) ( x 2 + 4 ) d x 0 x 2 + 2 x 2 + 1 x 2 + 4 d x int_(0)^(oo)(x^(2)+2)/((x^(2)+1)(x^(2)+4))dx\int_0^{\infty} \frac{x^2+2}{\left(x^2+1\right)\left(x^2+4\right)} d x0x2+2(x2+1)(x2+4)dx.
    b) sin 2 2 x 1 + x 2 d x sin 2 2 x 1 + x 2 d x int_(-oo)^(oo)(sin^(2)2x)/(1+x^(2))dx\int_{-\infty}^{\infty} \frac{\sin ^2 2 x}{1+x^2} d xsin22x1+x2dx.
\(a^2=b^2+c^2-2bc\:Cos\left(A\right)\)

MMT-005 Sample Solution 2024

 

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