RGU Question Paper Mathematics Semester VI 2020: Complex Analysis and Number Theory

 

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BSc (VI Semester) Examination

MATHEMATICS

Paper : MATH–361 (Old Course)

( Complex Analysis and Number Theory )

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Note : 1. Answer all questions.
2. The figures in the margin indicate full marks for the questions.

1. Answer any five parts : 2×5=10

(a) For any complex number $z$, find $\left|z\right|$, if $z-i{z}^2=0$.

(b) If $arg\left(z\right)=a$, then find $arg\left(iz\right)$.

(c) Evaluate
$\underset{C}{\overset{}{\int }}\frac{z}{(z+1)(z-3)}dz$
where $C$ is the circle $\left|z\right|=2$.

(d) State Cauchy’s inequality of complex integration.

(e) Let $a$, $b$ and $c$ be positive integers. If $a|b$ and $a|c$, then prove that $a|bx+cy$, where $x$ and $y$ are integers.

(f) Find the least positive integer $n$ such that $4^{31}\equiv n (mod 5)$.

(g) For $a$, $b$ and $c$ are positive integers, if $a|bc$ and $gcd(a,b)=1$, then prove that $a|c$.

(h) Let $n>2$ be an integer and $\mathrm{a }$be any other integer such that $gcd(a,n)=1$. Find the arithmetic inverse of $a$ modulo $n$.

(i) Find the cube roots of $z=1+i$.

2. Answer any four parts : 5×4=20

(a) Prove that $u=y^3-3x^{2}y$ is a harmonic function. Determine its harmonic conjugate and find the corresponding analytic function $f(z)$in $z$.

(b) If $f\left(z\right)$ is analytic on and inside a circle $C$ with centre at $z=a$, then show that the mean of the values of $f\left(z\right)$ on $C$ is $f\left(a\right)$.

(c) Let $z_1$ and $z_2$ be complex numbers. Prove that $\left|z_1+z_2\right|\le \left|z_1\right|+\left|z_2\right|$ and $\left|z_1-z_2\right|\ge \left|z_1\right|-\left|z_2\right|$.

(d) Prove that $ax\equiv ay(mod m)$ if and only if $x\equiv y(mod \frac{m}{d})$, where $d=gcd(a,m)$.

(e) Determine all solutions in positive integers of the Diophantine equation $x+3y=9$.

(f) If $gcd(a,b)=1$, then prove that $gcd(a+b,a-b)=1 or 2$.

(g) State and prove Gauss’ mean value theorem.

(h) Find the area of a triangle whose vertices are given by $z_1$, $z_2$ and $z_3$ in the complex plane.

3. Answer any five parts : 10×5=50

(a) (i) Show that the function $f\left(z\right)=\sqrt{\left|xy\right|}$ is not analytic at origin, although the Cauchy-Riemann equations are satisfied at origin. 5

(ii) Show that an analytic function with constant modulus is constant. 5

(b) (i) If $f\left(z\right)$ is analytic on and inside a simple closed curve and $z_0$ is a point interior to $C$, then prove that
$f\left(z_0\right)=\frac{1}{2\pi i}\underset{C}{\overset{}{\int }}\frac{f\left(z\right)}{(z-z_0)}d\mathrm{z \; }$6

(ii) Evaluate
$\underset{C}{\overset{}{\int }}\frac{e^{iz}}{z^3}dz$
where $C$ is the circle $\left|z\right|=2$. 4

(c) (i) Let $z_1$, $z_2$ and $z_3$ represent vertices of an equilateral triangle. Prove that
${z_1}^{2+}{z_2}^{2+}{z_3}^{2=}{z}_1{z}_2+z_2{z}_3+z_3{z}_1$ 5

(ii) Evaluate
$\underset{C}{\overset{}{\int }}\frac{z}{z^2+2}dz$
where $C$ is the circle $\left|z\right|=3$. What is the value of the integral if $C$ is the curve $\left|z\right|=1$? 5

(d) If $a$ and $b$ are any two integers not both zero, then prove that $gcd(a,b)$exists and is unique. 10

(e) (i) If $gcd(a,b)=d$, then prove that there exist integers $x$ and $y$ such that $d=ax+by$. 6

(ii) If $p$ is an odd prime and $\mathrm{a }$is any integer such that $gcd(a,p)=1$, then show that
$a^{(p-1)/2}\equiv 1 or -1(mod p)$. 4

(f) (i) Find all incongruent modulo 30 solutions of the linear congruence $9x\equiv 21(mod 30)$. 5

(ii) If for some integer $m>1, (m-1)!\equiv -1(mod m)$, then prove that $m$ must be a prime. 5

(g) (i) If $\varphi$ is the Euler’s phi-function and $m>0$ and $a$ are integers such that $gcd(a,m)=1$, then show that $a^{\varphi \left(m\right)}\equiv 1 (mod m)$. 6

(ii) State and prove Fermat’s little theorem. 4

(h) (i) If $f\left(z\right)$ and $\overline{f\left(z\right)}$ are analytic functions in a domain, then show that $f\left(z\right)$ is constant. 5

(ii) Show that
$\underset{z\to 0}{\lim }\frac{\overline{z}}{z}$
does not exist. 5

(i) State and prove Lionville’s theorem of complex integration. Give an example to verify the theorem. 10