Sequences and Series of Complex Numbers. Complex-Valued Functions of a Real Variable. Curves and Domains in the Complex Plane

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.

Determine for which real values of the parameter $\alpha$ the improper integral converges:

Prove that the improper integral $\int_{1}^{\infty } e^{i t^{\beta }} t^{-\alpha } d t$, for real values of the constants $\alpha$ and $\beta$, converges if and only if these constants are related by $\alpha > \min (1,1-\beta )$.

Let $f(t)$ be a complex-valued function, continuous on the segment $[a, b]$ and differentiable at each interior point of this segment. Prove that the number $\lambda = \frac{f(b)-f(a)}{b-a}$ belongs to the convex hull of the set of values taken by the function $f^{\prime }(t)$ on the interval $(a, b)$.

Let the function $f(t)$ satisfy the same conditions as in Problem 34, and let $g(t)$ be a real function, continuous on the segment $[a, b]$ and having a nonzero derivative at each interior point of this segment. Prove that the number $\lambda = \frac{f(b)-f(a)}{g(b)-g(a)}$ belongs to the convex hull of the set of values taken by the function $\frac{f^{\prime }(t)}{g^{\prime }(t)}$ on the interval $(a, b)$.

By considering the functions

on the segment $[0, \pi ]$, verify that the condition that the function $g(t)$ be real in Problem 35 is essential.

Let $f(t)$ and $g(t)$ be complex-valued functions, continuous on the segment $[a, b]$ and differentiable at each interior point of this segment. Prove that one can choose three points $\tau_{1}, \tau_{2}, \tau_{3}$ of the interval $(a, b)$ and three nonnegative numbers $\lambda_{1}, \lambda_{2}, \lambda_{3}$, for which $\lambda_{1}+\lambda_{2}+\lambda_{3} = 1$, in such a way that the following equality holds

Prove that $\lim_{t \rightarrow b-0} \frac{f(t)}{g(t)} = A$, if $f(t)$ and $g(t)$ are continuous complex-valued functions satisfying the conditions:

Determine which curve is defined by the parametric equation (indicate the set of points of the plane and the order in which they are traversed):

Let the curve $\gamma$ be given by the parametric equation $z = z(t), a \leqslant t \leqslant b$, and let the function $z(t)$ have at the point $t_{0}$ a nonzero derivative $z^{\prime }\left(t_{0}\right)$. Prove that the curve $\gamma$ has a tangent at the point $z\left(t_{0}\right)$ and that the complex number $z^{\prime }\left(t_{0}\right)$ represents, on the complex plane, a vector directed along this tangent.

Let the curve $\gamma$ be given by the parametric equation $z = z(t), a \leqslant t \leqslant b$, where $z(t)$ is a function having two continuous derivatives on the segment $[a, b]$. Denote by $\tau (t)$ the complex number representing the unit tangent vector to the curve $\gamma$ at the point $z(t)$ (directed the same way as the curve itself at this point); denote by $\nu (t)$ the complex number representing the unit normal vector to the curve $\gamma$ at the point $z(t)$ (directed to the right of the curve); denote by $\rho (t)$ the curvature of the curve $\gamma$ at the point $z(t)$. Prove the formulas:

Let the curve $\gamma$ be given by the parametric equation $z = z(t), a \leqslant t \leqslant b$, where $z(t)$ is a function continuously differentiable on the segment $[a, b]$. Prove that the curve $\gamma$ is rectifiable and that for its length $\lambda (\gamma )$ the formula holds

Prove that any piecewise-smooth curve is rectifiable.

Let $z = \varkappa (t), \quad 0 \leqslant t \leqslant l$, be the natural equation of the curve $\gamma$, and let the function $\varkappa (t)$ be twice continuously differentiable on the segment $[0, l]$, with $\tau (t), \nu (t), \rho (t)$ having the same meaning as in Problem 41. Prove the formulas:

Describe by means of inequalities the domain $D$ if its boundary $\partial D$ consists of one closed curve defined by the parametric equation:

Let $D$ be a bounded domain, and let its boundary $\partial D$ consist of one closed curve given by the parametric equation $z = z(t), a \leqslant t \leqslant b$, where the function $z(t)$ is continuously differentiable on the segment $[a, b]$. Prove that for the area $\sigma (D)$ of the domain $D$ the formula holds

Let $D$ be a bounded domain, and let its boundary $\partial D$ consist of $m$ closed curves given by the parametric equations

where $z_{k}(t)$ are functions continuously differentiable on the segments $\left[a_{k}, b_{k}\right]$, respectively. Prove that for the area $\sigma (D)$ of the domain $D$ the formula holds

Prove that for the domain $D$ to be convex it is necessary and sufficient that it be star-shaped with respect to each of its points.

Let the boundary $\partial D$ of a domain $D$ consist of one smooth closed curve with parametric equation $z = z(t), a \leqslant t \leqslant b$, where the function $z(t)$ has, on the segment $[a, b]$, a continuous and nonzero derivative. Prove that:

Prove that a domain $D$ star-shaped with respect to one of its points (in particular, a convex domain) is simply connected.