Introduction

Compute:

Write the complex number $z$ in trigonometric and exponential form:

Find all roots of the equation:

Solve the system of equations:

Solve the equation:

Let $z = z_{0}$ be a root of a polynomial $P(z)$ with real coefficients. Prove that $P\left(\bar{z}_{0}\right) = 0$, i.e., $\bar{z}_{0}$ is also a root of the polynomial $P(z)$.

Let $z_{1}$ and $z_{2}$ be fixed points of the complex plane. Give a geometric description of the set of all points $z$ satisfying the equation:

Let $\Delta_{1}$ be a triangle with vertices $z_{1}, z_{2}, z_{3}$, and let $\Delta_{2}$ be a triangle with vertices $w_{1}, w_{2}, w_{3}$. Prove that the triangle $\Delta_{1}$ is similar to the triangle $\Delta_{2}$ if

Determine which line in the plane is given by the equation:

Determine which set of points $z$ of the complex plane satisfies the inequality:

Let $A$ and $C$ be real, and let $B$ be a complex constant, and let $A C < \left|B\right|^{2}$. Prove that the equation

is the equation of a circle, and find the center and radius of this circle.

Prove that the equation of the circle passing through three given points $z_{1}, z_{2}, z_{3}$, not lying on one line, can be written in the form

Prove that for any positive value $K$ different from 1, the equation $\left|\frac{z-z_{1}}{z-z_{2}}\right| = K$ is the equation of a circle, and find the center and radius of this circle.

Prove that four pairwise distinct points $z_{1}, z_{2}, z_{3}, z_{4}$ lie on one circle (or on one line) if and only if the quantity $\frac{z_{2}-z_{1}}{z_{3}-z_{1}}: \frac{z_{2}-z_{4}}{z_{3}-z_{4}}$ is real.

Let $a$ be an arbitrary complex number satisfying the condition $\mathrm{Im}\left(a\right) > 0$. Prove that the quantity $\left|\frac{z-a}{z-\bar{a}}\right|$ is greater than unity in the lower half-plane, less than unity in the upper half-plane, and equal to unity on the real axis.

Let $a$ be an arbitrary real number. Prove that if the polynomial $P(z) = z^{n}+a_{1} z^{n-1}+\ldots +a_{n}$ has $n$ real roots, then the polynomial $Q(z) = P(z+i a)+P(z-i a)$ also has $n$ real roots.

Find, on the segment joining the points $z_{1}$ and $z_{2}$, the point that divides this segment in the ratio $\lambda_{1}: \lambda_{2}$, where $\lambda_{1}$ and $\lambda_{2}$ are positive numbers.

Prove that three pairwise distinct points $z_{1}, z_{2}, z_{3}$ lie on one line if and only if the quantity $\frac{z_{3}-z_{1}}{z_{2}-z_{1}}$ is real.

Prove that a point $\zeta$ lies on the segment joining the points $z_{1}$ and $z_{2}$ if and only if there exists a number $\alpha , 0 \leqslant \alpha \leqslant 1$, such that $\zeta = \alpha z_{1}+(1-\alpha ) z_{2}$.

Suppose that at the points $z_{1}, \ldots , z_{n}$ of the complex plane there are placed point masses with masses $\lambda_{1}, \ldots , \lambda_{n}$, respectively. Prove that the center of mass of such a system of point masses is located at the point $\zeta = \frac{\lambda_{1} z_{1}+\ldots +\lambda_{n} z_{n}}{\lambda_{1}+\ldots +\lambda_{n}}$.

Let the points $z_{1}, z_{2}, z_{3}$ lie on a circle centered at the point $z = 0$. Prove that the triangle with vertices at the points $z_{1}, z_{2}, z_{3}$ is equilateral if and only if $z_{1}+z_{2}+z_{3} = 0$.

Prove that points $z_{1}, z_{2}, z_{3}, z_{4}$ lying on one circle are the vertices of a rectangle if and only if $z_{1}+z_{3} = z_{2}+z_{4}$ (the points are numbered in the order in which they occur when traversing the circle).

Given three vertices $z_{1}, z_{2}, z_{3}$ of a parallelogram, numbered in the order in which they occur along its boundary. Find the fourth vertex $z_{4}$ of the parallelogram.

Prove that for any $z \in \mathbb {C}$ the following equality holds

Prove that for any $z_{1} \in \mathbb {C}, z_{2} \in \mathbb {C}$ the following equalities hold:

Prove that the quantity

is nonnegative for any $\lambda \in \mathbb {C}, \mu \in \mathbb {C}$ if and only if the conditions

Prove that for any $z_{k} \in \mathbb {C}, \zeta_{k} \in \mathbb {C}(k = 1,2, \ldots , n)$ the following inequality holds

(the Cauchy--Bunyakovsky--Schwarz inequality).

Prove that for any $z_{k} \in \mathbb {C}(k = 1,2, \ldots , n)$ the following inequality holds

Let $0 < s^{\prime } < s$. Prove that for any $z_{k} \in \mathbb {C}(k = 1,2, \ldots , n)$ the following inequality holds

Let $s > 0$. Prove that for any nonzero $z_{k} \in \mathbb {C} (k = 1,2, \ldots , n)$ the following inequality holds

Let $z_{1}, z_{2}, \ldots , z_{n}$ be arbitrary complex numbers. Prove that:

Let $p > 1, q > 1$, and $\frac{1}{p}+\frac{1}{q} = 1$. Prove that for any $z_{k} \in \mathbb {C}, \zeta_{k} \in \mathbb {C} (k = 1,2, \ldots , n)$ the following inequality holds

(the Hölder inequality).

Prove the convergence of the sequence and find its limit:

Prove that if $\left|z_{n}\right| \leqslant M < \infty$ for $n > n_{0}$, then from the sequence $\left\{ z_{n}\right\}$ one can select a subsequence $\left\{ z_{n_{k}}\right\}$ converging to a finite limit.

Let $\lim_{n \rightarrow \infty } z_{n} = A \neq \infty$. Prove that

Let $\lim_{n \rightarrow \infty } z_{n} = A \neq \infty , \lim_{n \rightarrow \infty } \zeta_{n} = B \neq \infty$. Prove that

Let the numbers $\lambda_{1}, \lambda_{2}, \ldots$ be positive, and let

Prove that from the equality $\lim_{n \rightarrow \infty } z_{n} = A \neq \infty$ it follows that

Prove that for the sequence $\left\{ z_{n}\right\}$ to converge to infinity it is necessary and sufficient that the sequence of real numbers $\left\{ \left|z_{n}\right|\right\}$ converge to $+\infty$.

Prove that $\lim_{n \rightarrow \infty } z_{n} = \infty$ if and only if $\lim_{n \rightarrow \infty } \frac{1}{z_{n}} = 0$.

Choose in space a coordinate system $\xi , \eta , \zeta$ such that the axes $O \xi$ and $O \eta$ coincide with the axes $O x$ and $O y$ of the complex plane, and the axis $O \zeta$ is directed along the diameter of the Riemann sphere (Fig. 2.2). Let $x = \mathrm{Re}\left(z\right)$, $y = \mathrm{Im}\left(z\right)$, and let the point $M(z)$ have spatial coordinates $(\xi , \eta , \zeta )$. Prove the formulas:

Fig. 2.2

Let the point $M(z)$ have spatial coordinates $(\xi , \eta , \zeta )$. Find the spatial coordinates of the point $M$:

Give a geometric description of the sets on the Riemann sphere corresponding to the following sets of the complex plane:

Prove that points $M\left(z_{1}\right)$ and $M\left(z_{2}\right)$ of the Riemann sphere, distinct from the points $O$ and $N$, are diametrically opposite if and only if the points $z_{1}$ and $z_{2}$ are related by the condition $z_{1} \bar{z}_{2} = -1$.

Prove that a circle on the Riemann sphere corresponds, on the complex plane, to either a circle or a line, and a line is obtained if and only if the circle on the Riemann sphere passes through its upper pole $N$.

Find the values of the parameter $a$ for which circles of the complex plane correspond to great circles on the Riemann sphere:

The distance in space between the points $M\left(z_{1}\right)$ and $M\left(z_{2}\right)$ is called the chordal distance between the points $z_{1}$ and $z_{2}$ of the extended complex plane and is denoted by the symbol $k\left(z_{1}, z_{2}\right)$. Prove the formulas:

Give a geometric description of the set of points $z$ of the complex plane satisfying the inequality:

Prove that the series $\sum_{n = 1}^{\infty } z_{n}$ converges absolutely if one of the following conditions holds:

Prove the absolute convergence of the series:

Let $\left\{ \lambda_{n}\right\}$ be a sequence of positive numbers such that $\lambda_{k+1} \leqslant \lambda_{k} \quad (k \in \mathbb {N}), \quad \lim_{n \rightarrow \infty } \lambda_{n} = 0$, and let $\left\{ z_{n}\right\}$ be a sequence of complex numbers such that $\left|\sum_{k = 1}^{n} z_{k}\right| \leqslant M, M > 0, n \in \mathbb {N}$. Prove that the series $\sum_{n = 1}^{\infty } \lambda_{n} z_{n}$ converges.

Find all values $\alpha \in \mathbb {R}$ for which the series converges:

Verify the differentiability of the following functions and find their derivatives:

Compute the integral:

Let the functions $z_{1}(t)$ and $z_{2}(t)$ be differentiable. Prove the formulas:

Let $\varphi (t)$ be a real function, differentiable at the point $t_{0}$, and let $z(t)$ be a complex-valued function, differentiable at the point $\varphi \left(t_{0}\right)$. Prove that the function $z_{1}(t) = z(\varphi (t))$ is differentiable at the point $t_{0}$ and that $z_{1}^{\prime }\left(t_{0}\right) = z^{\prime }\left(\varphi \left(t_{0}\right)\right) \varphi^{\prime }\left(t_{0}\right)$.

Let the function $z(t)$ be differentiable and nonzero. Prove the formulas:

Let the real function $\varphi (t)$ be monotone and continuously differentiable on the segment $[a, b]$, and let the complex-valued function $z(t)$ be continuous on the segment $[\varphi (a), \varphi (b)]$. Prove that

Let the complex-valued function $z(t)$ be continuous on the segment $a \leqslant t \leqslant b$. Prove the inequalities:

Let $z(t)$ and $\zeta (t)$ be complex-valued functions, continuous on the segment $a \leqslant t \leqslant b$. Prove that:

Let the complex-valued function $z(t)$ be continuous on the segment $[0,1]$ and let $z(0) \neq 0$. Prove that the improper integral $\int_{0}^{1} z(t) t^{-\alpha } d t$ converges for $\alpha < 1$ and diverges for $\alpha \geqslant 1$.

Let the complex-valued function $z(t)$ be continuous for $t \geqslant 1$ and let there exist a nonzero limit of this function as $t \rightarrow +\infty$. Prove that the improper integral $\int_{1}^{\infty } z(t) t^{-\alpha } d t$ converges for $\alpha > 1$ and diverges for $\alpha \leqslant 1$.

Let the complex-valued function $z(t)$ be continuous for $t \geqslant 1$ and let the functions $\mathrm{Re}\left(z(t)\right)$ and $\mathrm{Im}\left(z(t)\right)$ be nonnegative and monotone for $t \geqslant 1$. Prove that the improper integral $\int_{1}^{\infty } z(t) d t$ and the series $\sum_{n = 1}^{\infty } z(n)$ converge or diverge simultaneously.

Let $\varphi (t)$ be a real function, continuously differentiable for $t \geqslant 1$ and monotonically tending to zero as $t \rightarrow +\infty$, and let $z(t)$ be a complex-valued function, continuous for $t \geqslant 1$ and having the property that

Prove that the improper integral $\int_{1}^{\infty } z(t) \varphi (t) d t$ converges.