Complex Numbers and Number Sequences

The number $z + \overline{z}$ is real, $z - \overline{z}$ is purely imaginary, $z\overline{z}$ is real and not negative.

What sets of points in the $z$-plane are characterized by the conditions:

($a, b, \alpha , \beta , R, R'$ real, $z_{0}$ complex, $a < b$, $\alpha < \beta < \alpha + 2\pi$, $0 < R < R'$)?

What sets of points in the $z$-plane are characterized by the conditions

What open set of the $z$-plane is characterized by the condition

For what values of $R$ is this set connected, for what values of $R$ is it not connected?

Assume $\left|a\right| < 1$. For any point of the complex $z$-plane

is either $< 1$, or $= 1$, or $> 1$, and so the whole plane is divided into three subsets. Describe them.

Suppose $\mathrm{Re}\left(a\right) > 0$. For any point of the $z$-plane

is either $< 1$, or $= 1$, or $> 1$, and so the whole plane is divided into three subsets. Describe them.

Let $\alpha , \beta$ be real; $a$ complex; $\alpha , \beta$ and $a$ are fixed. Suppose that the complex variables $z_{1}$ and $z_{2}$ satisfy the relation

If $\alpha \beta - a\overline{a} < 0$ the points $\frac{z_{1}}{z_{2}}$ lie on a circle, possibly a line segment. (The left hand side of the equation is called a Hermitian form of the variables $z_{1}$ and $z_{2}$.)

Let $a$ and $b$ be positive constants and the real variable $t$ signify time. Describe the curves given by the three equations

Describe the motion of the point

where $a, b$ are positive constants and $t$ denotes time.

Let the radius vector $r$ and the argument $\vartheta$ be functions of the time. The complex function $z = re^{i\vartheta }$ of the real variable $t$ is represented by the motion of a point in a plane. Compute the components of velocity and of acceleration parallel and perpendicular to the radius vector.

For what values of $z$ is the absolute value of the $n$-th term $\frac{z^{n}}{n!}$ of

(exponential series in the complex plane) larger than the absolute value of any other term? $n = 0, 1, 2, \ldots$

For what values of $z$ is the absolute value of the $n$-th term of the series

(binomial series for $(1+t)^{z}$ with $t = 1$ and complex $z$) larger than the absolute value of any other term of this series? $n = 0, 1, 2, \ldots$

We put

For what values of $z$ is $\left|P_{n}(z)\right|$ larger than $\left|P_{0}(z)\right|, \left|P_{1}(z)\right|, \ldots , \left|P_{n-1}(z)\right|$, $\left|P_{n+1}(z)\right|, \ldots$? ($P_{n}(z)$ is the $n$-th partial product in the product expansion of $\frac{\sin \pi z}{\pi }$.)

We assume that the real functions $f(t)$ and $\varphi (t)$ are defined on the interval $a \leq t \leq b$, that $f(t)$ is positive and continuous and $\varphi (t)$ properly integrable. Then

Equality holds if and only if the function $\varphi (t)$ assumes the same value mod $2\pi$ at all its points of continuity.

Let $\varphi (t)$ be defined for $t \geq 0$ and be properly integrable over any finite interval. If

then

Equality holds if and only if $\varphi (t)$ assumes the same value mod $2\pi$ at all its points of continuity.

A polynomial of the form

where $p_{1} \geq 0, p_{2} \geq 0, \ldots , p_{n} \geq 0$, $p_{1} + p_{2} + \cdots + p_{n} > 0$, has just one positive zero.

If $z_{0}$ is a zero of the polynomial

then $\left|z_{0}\right|$ is not larger than the only positive zero $\zeta$ of the polynomial $z^{n} - \left|a_{1}\right|z^{n-1} - \left|a_{2}\right|z^{n-2} - \cdots - \left|a_{n}\right|$.

Assume $a_{n} \neq 0$. The absolute value of none of the zeros of the polynomial

is smaller than the only positive zero $\zeta$ of the polynomial

All the zeros of the polynomial $z^{n} + c$ are on the circle centred at $z = 0$ with radius $\left|c\right|^{1/n}$.

Let $c_{1}, c_{2}, \ldots , c_{n}$ be positive numbers and

The absolute values of the zeros of the polynomial

are not larger than

The absolute values of the roots of the equation

are not larger than the largest among the numbers

also they are not larger than the largest of the numbers

and they are certainly smaller than the largest of the numbers

Assume

The polynomial

cannot have a zero in the unit disc $\left|z\right| \leq 1$.

Suppose that all the coefficients $p_{1}, p_{2}, \ldots , p_{n}$ of the polynomial

are positive. Then the zeros of this polynomial lie in the annulus $\alpha \leq \left|z\right| \leq \beta$, where $\alpha$ is the smallest, $\beta$ the largest among the values

Let $a_{0}, a_{1}, a_{2}, \ldots , a_{n}$ be digits (in the ordinary decimal notation, that is integers between $0$ and $9$ inclusively) $n \geq 1$, $a_{n} \geq 1$. Then the zeros of the polynomial

are either in the open left half-plane or in the open disk

The best upper bound that may replace the last number is between $3$ and $4$.

We assume that all the zeros of the polynomial

are in the upper half-plane $\mathrm{Im}\left(z\right) > 0$. Let $a_{\nu } = \alpha_{\nu } + i\beta_{\nu }$, $\alpha_{\nu }, \beta_{\nu }$ real, $\nu = 0, 1, 2, \ldots , n$, and

Then the polynomials $U(z)$ and $V(z)$ have only real zeros.

Let $P(z) = 0$ stand for an algebraic equation of degree $n$ all the zeros of which are in the unit circle $\left|z\right| < 1$. Replacing each coefficient of $P(z)$ by its conjugate we obtain the polynomial $\overline{P}(z)$. We define $P^{*}(z) = z^{n}\overline{P}(z^{-1})$. The roots of the equation $P(z) + P^{*}(z) = 0$ are all on the unit circle $\left|z\right| = 1$.

[M. Fekete.] Define $\gamma = \lambda P(a) + \mu P(b)$, $0 < \lambda < 1$, $\lambda + \mu = 1$. If all the zeros of $P(z) - \gamma = a_{0}(z - z_{1})(z - z_{2})\cdots (z - z_{n})$ were outside the domain in question, the inequality

would hold, thus

If all the complex numbers $z_{1}, z_{2}, \ldots , z_{n}$ (considered as points in the complex plane) are on the same side of a straight line passing through the origin, then

Suppose $z_{1}, z_{2}, \ldots , z_{n}$ are arbitrary complex numbers that add up to zero. Any straight line $l$ through the origin separates the numbers $z_{1}, z_{2}, \ldots , z_{n}$ so that there are some $z_{\nu }$'s on each side of $l$ unless all the $z_{\nu }$'s lie on $l$ itself.

Let $z_{1}, z_{2}, \ldots , z_{n}$ be arbitrary points of the complex plane, $m_{1} > 0, m_{2} > 0, \ldots , m_{n} > 0$, $m_{1} + m_{2} + \cdots + m_{n} = 1$ and

Then there are points $z_{\nu }$ on both sides of any straight line through $z$ except when all the $z_{\nu }$'s lie on that straight line.

We can interpret the numbers $m_{1}, m_{2}, \ldots , m_{n}$ as masses fixed at the points $z_{1}, z_{2}, \ldots , z_{n}$. Then the point $z$ defined above is the center of gravity of this mass distribution. If we consider all such mass distributions at the points $z_{1}, z_{2}, \ldots , z_{n}$ the corresponding centers of gravity cover the interior of a convex polygon, the smallest one containing the points $z_{1}, z_{2}, \ldots , z_{n}$. The only exception arises when all the points are on a straight line. Then the centers of gravity fill out the interior of the smallest line segment that contains all the points $z_{1}, z_{2}, \ldots , z_{n}$.

The derivative $P'(z)$ of $P(z)$ cannot have any zeros outside the smallest convex polygon that contains all the zeros of $P(z)$ (considered as points in the complex plane). Those zeros of $P'(z)$ that are not zeros of $P(z)$ are, in fact, equilibrium positions of a certain electrostatic field.

Let $z_{1}, z_{2}, \ldots , z_{n}$ be arbitrary complex numbers, $z_{\mu } \neq z_{\nu }$ for all $\mu \neq \nu$, $\mu , \nu = 1, 2, \ldots , n$. We consider all the polynomials $P(z)$ that vanish only at the points $z_{1}, z_{2}, \ldots , z_{n}$ (having there zeros of arbitrary order). The set of the zeros of the derivatives $P'(z)$ of all these polynomials is everywhere dense in the smallest convex polygon that contains $z_{1}, z_{2}, \ldots , z_{n}$.

Let $\varrho_{1}, \varrho_{2}, \ldots , \varrho_{p}$ be positive numbers, $a_{1}, a_{2}, \ldots , a_{p}$ arbitrary complex numbers, and let the polynomials $A(z)$ and $B(z)$ of degree $p$ and $p - 1$ resp. be related by

Suppose that the polynomial $P(z)$ is a divisor of $A(z)P''(z) + 2B(z)P'(z)$, i.e.

where $C(z)$ denotes a polynomial. Then the zeros of $P(z)$ lie in the smallest convex polygon that contains the numbers $a_{1}, a_{2}, \ldots , a_{p}$.

Let $\varrho_{1}, \varrho_{2}, \ldots , \varrho_{p}$ be positive numbers, $a_{1}, a_{2}, \ldots , a_{p}$ arbitrary complex numbers, $A(z)$ and $B(z)$ related as in problem 33, and $z_{\nu }$, $\nu = 1, \ldots , n$ the zeros of a polynomial $P(z)$ with $A(z_{\nu }) \neq 0$. Then the zeros of the polynomial $A(z)P''(z) + 2B(z)P'(z)$ lie in the smallest convex polygon that contains $z_{1}, z_{2}, \ldots , z_{n}, a_{1}, a_{2}, \ldots , a_{p}$.

If a polynomial $f(z)$ whose coefficients are all real has only real zeros then this is true also for its derivative $f'(z)$. If $f(z)$ has complex zeros then they appear in pairs, the two zeros forming a pair are mirror images to each other with respect to the real axis; they are complex conjugates. We draw all those disks the ``vertical'' diameters of which are the line segments connecting the conjugate zeros of such pairs. If $f'(z)$ has any complex zeros they lie in these disks.

Assume that the numbers $z_{1}, z_{2}, \ldots , z_{n}, \ldots$ are all in the sector $-\alpha \leq \arg z \leq \alpha$, $\alpha < \frac{\pi }{2}$. Then the series

are either both convergent or both divergent.

Suppose that the numbers $z_{1}, z_{2}, \ldots , z_{n}, \ldots$ are all in the half-plane $\mathrm{Re}\left(z\right) \geq 0$ and that the two series

converge. Then $\left|z_{1}\right|^{2} + \left|z_{2}\right|^{2} + \cdots + \left|z_{n}\right|^{2} + \cdots$ converges too.

There exist complex sequences $z_{1}, z_{2}, \ldots , z_{n}, \ldots$ for which all the series

converge and all the series

diverge.

Let $z_{1}, z_{2}, \ldots , z_{n}, \ldots$ be arbitrary complex numbers. If there exists a positive distance $\delta$ such that $\left|z_{l} - z_{k}\right| \geq \delta$ for $l < k$, $l, k = 1, 2, 3, \ldots$ the convergence exponent of the sequence $\left|z_{1}\right|, \left|z_{2}\right|, \left|z_{3}\right|, \ldots$ is at most $2$. [Problem 114 of Part I.]

The limit points of the complex numbers

fill out the entire circle with radius $(1+\alpha^{2})^{-1/2}$ and center at the origin.

Find the locus of the limit points of the complex sequence $z_{1}, z_{2}, \ldots , z_{n}, \ldots$, where

Put

and connect the points $z_{n-1}$ and $z_{n}$ by a straight line. The distance between these two points is always $1$. The polygonal line connecting the successive points approaches with increasing $n$ an Archimedean spiral; if $z_{n} = r_{n}e^{i\varphi_{n}}$, $r_{n} > 0$, $0 < \varphi_{n} - \varphi_{n-1} < \frac{\pi }{2}$, then

Let $t$ be a fixed real number and put $z = 2ne^{\frac{it}{\sqrt{n}}}$. Then

Prove the easier part of the above mentioned proposition: If the conditions (1) and (2) are satisfied the array preserves convergence. [Problems 66 and 80 of Part I.]

What conditions must the series $u_{0} + u_{1} + u_{2} + \cdots + u_{n} + \cdots$ satisfy in order that its Cauchy product [Problem 34 of Part I, Problem 23 of Part II, Part VIII Chapter 1 5] with any convergent series $v_{0} + v_{1} + v_{2} + \cdots + v_{n} + \cdots$ results in a convergent series

What condition must the series $u_{1}+u_{2}+\cdots +u_{n}+\cdots$ satisfy in order that its Dirichlet product [Part VIII, Chapter 1, 5] with any convergent series $v_{1}+v_{2}+\cdots +v_{n}+\cdots$ results in a convergent series

The sequence of factors $\gamma_{0}, \gamma_{1}, \gamma_{2}, \ldots , \gamma_{n}, \ldots$ turns any convergent series $a_{0}+a_{1}+a_{2}+\cdots +a_{n}+\cdots$ into a convergent series

if and only if the series

converges.

[Cf.G. Polya, Problem: Arch. Math. Phys. Ser. 3, Vol. 24, p.282 (1916). Solved by S. Sidon: Arch. Math. Phys. Ser. 3, Vol. 26, p.68 (1917).] Set

The existence of $\lim_{n\to \infty }(u_{1}+u_{2}+\cdots +u_{n-1}+cu_{n}) = \alpha$ implies the existence of $\lim_{n\to \infty }(u_{1}+u_{2}+\cdots +u_{n-1}+u_{n}) = \alpha$ in two cases only: if $c = 0$ or if $\mathrm{Re}\left(c\right) > \frac{1}{2}$, but not if $\mathrm{Re}\left(c\right) \leq \frac{1}{2}$, $c \neq 0$.

Let $u_{0}, u_{1}, u_{2}, \ldots , u_{n}, \ldots$ be arbitrary complex numbers. For what values of $c$ does the existence of

imply the existence of $\lim_{n\to \infty } u_{n}$?

If the Dirichlet series

converges for $s = \sigma + i\tau$, $\sigma , \tau$ real, $\sigma > 0$, then

If every subseries of a series with complex terms converges the series converges absolutely.

Assume that the series $\left|z_{1}\right| + \left|z_{2}\right| + \cdots + \left|z_{n}\right| + \cdots$ diverges. Then there exists a direction of accumulation, that is a real number $\alpha$ such that those terms of the series $z_{1}+z_{2}+\cdots +z_{n}+\cdots$ that are contained in the sector $\alpha - \varepsilon < \arg z < \alpha + \varepsilon$ constitute an absolutely divergent subseries for any $\varepsilon > 0$.

If $\lim_{n\to \infty } z_{n} = 0$ and if the positive real axis is a direction of accumulation of the conditionally convergent series $z_{1}+z_{2}+z_{3}+\cdots$ then there exists a subseries $z_{r_{1}}+z_{r_{2}}+z_{r_{3}}+\cdots$ the real part of which diverges to $+\infty$ and the imaginary part converges to a finite number.

If the series $z_{1}+z_{2}+z_{3}+\cdots$ is convergent, but not absolutely convergent, any value represented by a point of a certain straight line can be obtained as the sum of the series rearranged in a suitable order.