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+ib$$z=a+ib$z=a+ibz=a+i b$z=a+ib$, where $a$$a$aa$a$ and $b$$b$bb$b$ are integers, then $|1+z+z^2+\cdots +z^n|\ge |z{|}^n$$\left|1+z+z^2+\cdots +z^n\right|\ge |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+z^2+\cdots +z^n|\ge |z{|}^n$ if $a>0$$a>0$a > 0a>0$a>0$.
   ii) If $f(z)$$f(z)$f(z)f(z)$f(z)$ and $\overline{f(z)}$$\overline{f(z)}$bar(f(z))\overline{f(z)}$\overline{f(z)}$ are analytic functions in $a$$a$aa$a$ domain, then $f$$f$ff$f$ is necessarily a constant.
   iii) A real-valued function $u(x,y)$$u(x,y)$u(x,y)u(x, y)$u(x,y)$ is harmonic in $D$$D$DD$D$ iff $u(x,-y)$$u(x,-y)$u(x,-y)u(x,-y)$u(x,-y)$ is harmonic in $D$$D$DD$D$.
   iv) $\underset{n\to \mathrm{\infty }}{lim}(n!{)}^{1/n}=\mathrm{\infty }$$\underset{n\to \mathrm{\infty }}{lim} (n!{)}^{1/n}=\mathrm{\infty }$lim_(n rarr oo)(n!)^(1//n)=oo\lim _{n \rightarrow \infty}(n !)^{1 / n}=\infty$\underset{n\to \mathrm{\infty }}{lim}(n!{)}^{1/n}=\mathrm{\infty }$.
   v) The inequality $|e^a-e^b|\le |a-b|$$\left|e^a-e^b\right|\le |a-b|$|e^(a)-e^(b)| <= |a-b|\left|e^a-e^b\right| \leq|a-b|$|e^a-e^b|\le |a-b|$ holds for $a,b\in D=\{w:\mathrm{Re}w\le 0\}$$a,b\in D=\{w:\mathrm{Re}w\le 0\}$a,b in D={w:Re w <= 0}a, b \in D=\{w: \operatorname{Re} w \leq 0\}$a,b\in D=\{w:\mathrm{Re}w\le 0\}$.
   vi) If $f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n(z-a{)}^n$$f(z)=\sum _{n=0}^{\mathrm{\infty }} a_n(z-a{)}^n$f(z)=sum_(n=0)^(oo)a_(n)(z-a)^(n)f(z)=\sum_{n=0}^{\infty} a_n(z-a)^n$f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n(z-a{)}^n$ has the property that $\sum _{n=0}^{\mathrm{\infty }}{f}^{(n)}(a)$$\sum _{n=0}^{\mathrm{\infty }} f^{(n)}(a)$sum_(n=0)^(oo)f^((n))(a)\sum_{n=0}^{\infty} f^{(n)}(a)$\sum _{n=0}^{\mathrm{\infty }}{f}^{(n)}(a)$ converges, then $f$$f$ff$f$ is necessarily an entire function.
   vii) If a power series $\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n$$\sum _{n=0}^{\mathrm{\infty }} a_n{z}^n$sum_(n=0)^(oo)a_(n)z^(n)\sum_{n=0}^{\infty} a_n z^n$\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ converges for $|z|<1$$|z|<1$|z| < 1|z|<1$|z|<1$ and if $b_n\in \mathbb{C}$$b_n\in \mathbb{C}$b_(n)inCb_n \in \mathbb{C}$b_n\in \mathbb{C}$ is such that $\left|b_n\right|<n^2\left|a_n\right|$$\left|b_n\right|<n^2\left|a_n\right|$|b_(n)| < n^(2)|a_(n)|\left|b_n\right|<n^2\left|a_n\right|$\left|b_n\right|<n^2\left|a_n\right|$ for all $n\ge 0$$n\ge 0$n >= 0n \geq 0$n\ge 0$, then $\sum _{n=0}^{\mathrm{\infty }}{b}_n{z}^n$$\sum _{n=0}^{\mathrm{\infty }} b_n{z}^n$sum_(n=0)^(oo)b_(n)z^(n)\sum_{n=0}^{\infty} b_n z^n$\sum _{n=0}^{\mathrm{\infty }}{b}_n{z}^n$ converges for $|z|<1$$|z|<1$|z| < 1|z|<1$|z|<1$.
   viii) If $f$$f$ff$f$ is entire and $f(z)=f(-z)$$f(z)=f(-z)$f(z)=f(-z)f(z)=f(-z)$f(z)=f(-z)$ for all $z$$z$zz$z$, then there exists an entire function $g$$g$gg$g$ such that $f(z)=g\left(z^2\right)$$f(z)=g\left(z^2\right)$f(z)=g(z^(2))f(z)=g\left(z^2\right)$f(z)=g\left(z^2\right)$ for all $z\in \mathbb{C}$$z\in \mathbb{C}$z inCz \in \mathbb{C}$z\in \mathbb{C}$.
   ix) A mobius transformation which maps the upper half plane $\{z:\mathrm{Im}z>0\}$$\{z:\mathrm{Im}z>0\}${z:Im z > 0}\{z: \operatorname{Im} z>0\}$\{z:\mathrm{Im}z>0\}$ onto itself and fixing $0,\mathrm{\infty }$$0,\mathrm{\infty }$0,oo0, \infty$0,\mathrm{\infty }$ and no other points, must be of the form $Tz=\alpha z$$Tz=\alpha z$Tz=alpha zT z=\alpha z$Tz=\alpha z$ for some $\alpha >0$$\alpha >0$alpha > 0\alpha>0$\alpha >0$ and $\alpha \ne 1$$\alpha \ne 1$alpha!=1\alpha \neq 1$\alpha \ne 1$.
   x) If $f$$f$ff$f$ is entire and $\mathrm{Re}f(z)$$\mathrm{Re}f(z)$Re f(z)\operatorname{Re} f(z)$\mathrm{Re}f(z)$ is bounded as $|z|\to \mathrm{\infty }$$|z|\to \mathrm{\infty }$|z|rarr oo|z| \rightarrow \infty$|z|\to \mathrm{\infty }$, then $f$$f$ff$f$ is constant.
2. a) If $f=u+iv$$f=u+iv$f=u+ivf=u+i v$f=u+iv$ is entire such that $u_x+v_y=0$$u_x+v_y=0$u_(x)+v_(y)=0u_x+v_y=0$u_x+v_y=0$ in $\mathbb{C}$$\mathbb{C}$C\mathbb{C}$\mathbb{C}$ then show that $f$$f$ff$f$ has the form $f(z)=az+b$$f(z)=az+b$f(z)=az+bf(z)=a z+b$f(z)=az+b$ where $a,\mathbf{b}$$a,\mathbf{b}$a,ba, \mathbf{b}$a,\mathbf{b}$ are constants with $\mathrm{Re}a=0$$\mathrm{Re}a=0$Re a=0\operatorname{Re} a=0$\mathrm{Re}a=0$.
   b) Consider $f(z)=z^2-z$$f(z)=z^2-z$f(z)=z^(2)-zf(z)=z^2-z$f(z)=z^2-z$ and the closed circular region $R=\{z:|z|\le 1\}$$R=\{z:|z|\le 1\}$R={z:|z| <= 1}R=\{z:|z| \leq 1\}$R=\{z:|z|\le 1\}$. Find points in $R$$R$RR$R$ where $|f(z)|$$|f(z)|$|f(z)||f(z)|$|f(z)|$ has its maximum and minimum values.
   c) Find the points where the function $f(z)=\frac{\log (z+4)}{z^2+i}$$f(z)=\frac{\log (z+4)}{z^2+i}$f(z)=(log(z+4))/(z^(2)+i)f(z)=\frac{\log (z+4)}{z^2+i}$f(z)=\frac{\log (z+4)}{z^2+i}$ is not analytic.
3. a) Evaluate the following integrals:
   i) $I={\int }_0^{2\pi }f\left(e^{i\theta }\right){\cos }^2(\theta /2)d\theta$$I={\int }_0^{2\pi } f\left(e^{i\theta }\right){\cos }^2(\theta /2)d\theta$I=int_(0)^(2pi)f(e^(i theta))cos^(2)(theta//2)d thetaI=\int_0^{2 \pi} f\left(e^{i \theta}\right) \cos ^2(\theta / 2) d \theta$I={\int }_0^{2\pi }f\left(e^{i\theta }\right){\cos }^2(\theta /2)d\theta$.
   ii) $I={\int }_0^{2\pi }f\left(e^{i\theta }\right){\sin }^2\theta /2d\theta$$I={\int }_0^{2\pi } f\left(e^{i\theta }\right){\sin }^2\theta /2d\theta$quad I=int_(0)^(2pi)f(e^(i theta))sin^(2)theta//2d theta\quad I=\int_0^{2 \pi} f\left(e^{i \theta}\right) \sin ^2 \theta / 2 d \theta$I={\int }_0^{2\pi }f\left(e^{i\theta }\right){\sin }^2\theta /2d\theta$.
   b) Find the image of the circle $|z|=r(r\ne 1)$$|z|=r(r\ne 1)$|z|=r(r!=1)|z|=r(r \neq 1)$|z|=r(r\ne 1)$ under the mapping $w=f(z)=\frac{z-i}{z+i}$$w=f(z)=\frac{z-i}{z+i}$w=f(z)=(z-i)/(z+i)w=f(z)=\frac{z-i}{z+i}$w=f(z)=\frac{z-i}{z+i}$. What happens when $r=1$$r=1$r=1r=1$r=1$ ?
4. a) If $p(z)=a_0+a_{1}z+\cdots +a_{n-1}{z}^{n-1}+z^n(n\ge 1)$$p(z)=a_0+a_{1}z+\cdots +a_{n-1}{z}^{n-1}+z^n(n\ge 1)$p(z)=a_(0)+a_(1)z+cdots+a_(n-1)z^(n-1)+z^(n)(n >= 1)p(z)=a_0+a_1 z+\cdots+a_{n-1} z^{n-1}+z^n(n \geq 1)$p(z)=a_0+a_{1}z+\cdots +a_{n-1}{z}^{n-1}+z^n(n\ge 1)$, then show that there exists a real $R>0$$R>0$R > 0R>0$R>0$ such that $2^{-1}|z{|}^n\le |p(z)|\le 2|z{|}^n$$2^{-1}|z{|}^n\le |p(z)|\le 2|z{|}^n$2^(-1)|z|^(n) <= |p(z)| <= 2|z|^(n)2^{-1}|z|^n \leq|p(z)| \leq 2|z|^n$2^{-1}|z{|}^n\le |p(z)|\le 2|z{|}^n$ for $|z|\ge R$$|z|\ge R$|z| >= R|z| \geq R$|z|\ge R$.
   b) Find all solutions to the equation $\sin z=5$$\sin z=5$sin z=5\sin z=5$\sin z=5$.
5. a) Find the constant $c$$c$cc$c$ such that $f(z)=\frac{1}{z^n+z^{n-1}+\cdots +z^2+z^{-n}}+\frac{c}{z-1}$$f(z)=\frac{1}{z^n+z^{n-1}+\cdots +z^2+z^{-n}}+\frac{c}{z-1}$f(z)=(1)/(z^(n)+z^(n-1)+cdots+z^(2)+z^(-n))+(c)/(z-1)f(z)=\frac{1}{z^n+z^{n-1}+\cdots+z^2+z^{-n}}+\frac{c}{z-1}$f(z)=\frac{1}{z^n+z^{n-1}+\cdots +z^2+z^{-n}}+\frac{c}{z-1}$ can be extended to be analytic at $z=1$$z=1$z=1z=1$z=1$, when $n\in \mathbb{N}$$n\in \mathbb{N}$n inNn \in \mathbb{N}$n\in \mathbb{N}$ is fixed.
   b) Find all the singularities of the function $f(z)=\exp \left(\frac{z}{\sin z}\right)$$f(z)=\exp \left(\frac{z}{\sin z}\right)$f(z)=exp((z)/(sin z))f(z)=\exp \left(\frac{z}{\sin z}\right)$f(z)=\exp \left(\frac{z}{\sin z}\right)$.
   c) Evaluate ${\text{∮}}_C\frac{dz}{z^2+1}$${\text{∮}}_C\frac{dz}{z^2+1}$oint_(C)(dz)/(z^(2)+1)\oint_C \frac{d z}{z^2+1}${\text{∮}}_C\frac{dz}{z^2+1}$ where $c$$c$cc$c$ is the circle $|z|=4$$|z|=4$|z|=4|z|=4$|z|=4$.
6. a) Find the maximum modulus of $f(z)=2z+5i$$f(z)=2z+5i$f(z)=2z+5if(z)=2 z+5 i$f(z)=2z+5i$ on the closed circular region defined by $|z|\le 2$$|z|\le 2$|z| <= 2|z| \leq 2$|z|\le 2$.
   b) Evaluate ${\int }_C\frac{z^3+3}{z(z-i{)}^2}dz$${\int }_C \frac{z^3+3}{z(z-i{)}^2}dz$int _(C)(z^(3)+3)/(z(z-i)^(2))dz\int_C \frac{z^3+3}{z(z-i)^2} d z${\int }_C\frac{z^3+3}{z(z-i{)}^2}dz$, where $c$$c$cc$c$ is the eight like figure shown in Fig. 1.
   

   

   
   c) Find the radius of convergence of the following series.
   i) $\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}}{k!}(z-1-i{)}^k$$\sum _{k=1}^{\mathrm{\infty }} \frac{(-1{)}^{k+1}}{k!}(z-1-i{)}^k$quadsum_(k=1)^(oo)((-1)^(k+1))/(k!)(z-1-i)^(k)\quad \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k !}(z-1-i)^k$\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}}{k!}(z-1-i{)}^k$
   ii) $\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{6k+1}{2k+5}\right)}^k(z-2i{)}^k$$\sum _{k=1}^{\mathrm{\infty }} {\left(\frac{6k+1}{2k+5}\right)}^k(z-2i{)}^k$quadsum_(k=1)^(oo)((6k+1)/(2k+5))^(k)(z-2i)^(k)\quad \sum_{k=1}^{\infty}\left(\frac{6 k+1}{2 k+5}\right)^k(z-2 i)^k$\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{6k+1}{2k+5}\right)}^k(z-2i{)}^k$
7. a) Expand $f(z)=\frac{1}{(z-1{)}^2(z-3)}$$f(z)=\frac{1}{(z-1{)}^2(z-3)}$f(z)=(1)/((z-1)^(2)(z-3))f(z)=\frac{1}{(z-1)^2(z-3)}$f(z)=\frac{1}{(z-1{)}^2(z-3)}$ in a Laurent series valid for
   i) $0<|z-1|<2$$0<|z-1|<2$0 < |z-1| < 20<|z-1|<2$0<|z-1|<2$ and
   ii) $0<|z-3|<2$$0<|z-3|<2$quad0 < |z-3| < 2\quad 0<|z-3|<2$0<|z-3|<2$.
   b) Find the zeros and singularities of the function $f(z)=\frac{z}{4{\cos }^{2}z-1}$$f(z)=\frac{z}{4{\cos }^{2}z-1}$f(z)=(z)/(4cos^(2)z-1)f(z)=\frac{z}{4 \cos ^2 z-1}$f(z)=\frac{z}{4{\cos }^{2}z-1}$ in $|z|\le 1$$|z|\le 1$|z| <= 1|z| \leq 1$|z|\le 1$. Also find the residue at the poles.
   c) Prove that the linear fractional transformation $\varphi (z)=\frac{2z-1}{2-z}$$\varphi (z)=\frac{2z-1}{2-z}$phi(z)=(2z-1)/(2-z)\phi(z)=\frac{2 z-1}{2-z}$\varphi (z)=\frac{2z-1}{2-z}$ maps the circle $c:|z|=1$$c:|z|=1$c:|z|=1c:|z|=1$c:|z|=1$ into itself. Also prove that $f(z)$$f(z)$f(z)f(z)$f(z)$ is conformal in $\overline{D}=\{z:|z|\le 1\}$$\overline{D}=\{z:|z|\le 1\}$bar(D)={z:|z| <= 1}\bar{D}=\{z:|z| \leq 1\}$\overline{D}=\{z:|z|\le 1\}$.
8. a) Find the image of the semi-infinite strip $x>0,0<y<1$$x>0,0<y<1$x > 0,0 < y < 1x>0,0<y<1$x>0,0<y<1$ when $w=i/z$$w=i/z$w=i//zw=i / z$w=i/z$. Sketch the strip and its image.
   b) Show that there is only one linear fractional transformation that maps three given distinct points $z_1,z_2$$z_1,z_2$z_(1),z_(2)z_1, z_2$z_1,z_2$ and $z_3$$z_3$z_(3)z_3$z_3$ in the extended $z$$z$zz$z$ plane onto three specified distinct points $w_1,w_2$$w_1,w_2$w_(1),w_(2)w_1, w_2$w_1,w_2$ and $w_3$$w_3$w_(3)w_3$w_3$ in the extended $w$$w$ww$w$ plane.
9. Evaluate the following integrals
   a) ${\int }_0^{\mathrm{\infty }}\frac{x^2+2}{(x^2+1)(x^2+4)}dx$${\int }_0^{\mathrm{\infty }} \frac{x^2+2}{\left(x^2+1\right)\left(x^2+4\right)}dx$int_(0)^(oo)(x^(2)+2)/((x^(2)+1)(x^(2)+4))dx\int_0^{\infty} \frac{x^2+2}{\left(x^2+1\right)\left(x^2+4\right)} d x${\int }_0^{\mathrm{\infty }}\frac{x^2+2}{(x^2+1)(x^2+4)}dx$.
   b) ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\sin }^{2}2x}{1+x^2}dx$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} \frac{{\sin }^{2}2x}{1+x^2}dx$int_(-oo)^(oo)(sin^(2)2x)/(1+x^(2))dx\int_{-\infty}^{\infty} \frac{\sin ^2 2 x}{1+x^2} d x${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\sin }^{2}2x}{1+x^2}dx$.