Zeros and Singularities

Let $\mathbb {C}^{3}$ denote the set of ordered triples of complex numbers. Define a map $F : \mathbb {C}^{3} \to \mathbb {C}^{3}$ by

Prove that $F$ is onto but not one-to-one.

Prove that the polynomial

has at least one root in the disc $\left|z\right| < 1$.

Suppose that $f$ is analytic inside and on the unit circle $\left|z\right| = 1$ and satisfies $\left|f(z)\right| < 1$ for $\left|z\right| = 1$. Show that the equation $f(z) = z^{3}$ has exactly three solutions (counting multiplicities) inside the unit circle.

How many zeros does the complex polynomial

have in the annulus $1 < \left|z\right| < 2$?

Consider the polynomial

How many zeros (counting multiplicities) does $p$ have in the annular region $1 < \left|z\right| < 2$?

Find the number of roots of

which lie between the two circles $\left|z\right| = 1$ and $\left|z\right| = 2$.

Let $\varepsilon$ be a positive number. Prove that the polynomial $p(z) = \varepsilon z^{3} - z^{2} - 1$ has exactly two roots in the half-plane $\mathrm{Re}\left(z\right) < 0$.

Let $r < 1 < R$. Show that for all sufficiently small $\varepsilon > 0$, the polynomial

has exactly five roots (counted with their multiplicities) inside the annulus

Let the $3 \times 3$ matrix function $A$ be defined on the complex plane by

How many distinct values of $z$ are there such that $\left|z\right| < 1$ and $A(z)$ is not invertible?

How many roots does the polynomial $z^{4} + 3z^{2} + z + 1$ have in the right half $z$-plane?

Prove that if the nonconstant polynomial $p(z)$, with complex coefficients, has all of its roots in the half-plane $\mathrm{Re}\left(z\right) > 0$, then all of the roots of its derivative are in the same half-plane.

Let $f$ be a nonconstant entire function whose values on the real axis are real and nonnegative. Prove that all real zeros of $f$ have even order.

Let $p$ be a nonconstant polynomial with real coefficients and only real roots. Prove that for each real number $r$, the polynomial $p - rp'$ has only real roots.

Prove that if $1 < \lambda < \infty$, the function

has only one zero in the half-plane $\mathrm{Re}\left(z\right) < 0$, and that this zero is real.

Prove that for every $\lambda > 1$, the equation $ze^{\lambda -z} = 1$ has exactly one root in the disc $\left|z\right| < 1$ and that this root is real.

Prove that for any $a \in \mathbb {C}$ and any integer $n \geq 2$, the equation $1 + z + az^{n} = 0$ has at least one root in the disc $\left|z\right| \leq 2$.

Prove that the polynomial $z^{4} + z^{3} + 1$ has exactly one root in the quadrant $\left\{ z = x+iy \mid x,y > 0\right\}$.

Let $f$ be analytic in an open set containing the closed unit disc. Suppose that $\left|f(z)\right| > m$ for $\left|z\right| = 1$ and $\left|f(0)\right| < m$. Prove that $f(z)$ has at least one zero in the open unit disc $\left|z\right| < 1$.

Let $0 < a_{0} \leq a_{1} \leq \cdots \leq a_{n}$. Prove that the equation

has no roots in the disc $\left|z\right| < 1$.

Show that the polynomial $p(z) = z^{5} - 6z + 3$ has five distinct complex roots, of which exactly three (and not five) are real.

Let $c_{0}, c_{1}, \ldots , c_{n-1}$ be complex numbers. Prove that all the zeros of the polynomial

lie in the open disc with center $0$ and radius

Let $P(x)$ be a polynomial with real coefficients and with leading coefficient $1$. Suppose that $P(0) = -1$ and that $P(x)$ has no complex zeros inside the unit circle. Prove that $P(1) = 0$.

Prove that the number of roots of the equation $z^{2n} + \alpha^{2}z^{2n-1} + \beta^{2} = 0$ ($n$ a natural number, $\alpha$ and $\beta$ real, nonzero) that have positive real part is

Let $\rho > 0$. Show that for $n$ large enough, all the zeros of

lie in the circle $\left|z\right| < \rho$.

Do the functions $f(z) = e^{z} + z$ and $g(z) = ze^{z} + 1$ have the same number of zeros in the strip $-\frac{\pi }{2} < \mathrm{Im}\left(z\right) < \frac{\pi }{2}$?

Let $a$ be a complex number and $\varepsilon$ a positive number. Prove that the function $f(z) = \sin z + \dfrac {1}{z-a}$ has infinitely many zeros in the strip $\left|\mathrm{Im}\left(z\right)\right| < \varepsilon$.

Suppose that $f$ is holomorphic on some neighborhood of $a$ in the complex plane. Prove that either $f$ is constant on some neighborhood of $a$, or there exist an integer $n > 0$ and real numbers $\delta , \varepsilon > 0$ such that for each complex number $b$ satisfying $0 < \left|b-f(a)\right| < \varepsilon$, the equation $f(z) = b$ has exactly $n$ roots in $\left\{ z \in \mathbb {C} \mid \left|z-a\right| < \delta \right\}$.

Let $\hat{a}_{0} + \hat{a}_{1}z + \cdots + \hat{a}_{n}z^{n}$ be a polynomial having $\hat{z}$ as a simple root. Show that there is a continuous function $r : U \to \mathbb {C}$, where $U$ is a neighborhood of $(\hat{a}_{0}, \ldots , \hat{a}_{n})$ in $\mathbb {C}^{n+1}$, such that $r(a_{0}, \ldots , a_{n})$ is always a root of $a_{0} + a_{1}z + \cdots + a_{n}z^{n}$, and $r(\hat{a}_{0}, \ldots , \hat{a}_{n}) = \hat{z}$.

Let

where all the $a_{n}$ are nonnegative reals, and the series has radius of convergence $1$. Prove that $f(z)$ cannot be analytically continued to a function analytic in a neighborhood of $z = 1$.

Let $f$ be a meromorphic function on $\mathbb {C}$ which is analytic in a neighborhood of $0$. Let its Maclaurin series be

with all $a_{k} \geq 0$. Suppose there is a pole of modulus $r > 0$ and no pole has modulus $< r$. Prove there is a pole at $z = r$.

Decide, without too much computation, whether a finite limit

exists, where $z$ is a complex variable, and if yes, compute the limit.

Let $f$ and $g$ be entire functions such that $\lim_{z \to \infty } f(g(z)) = \infty$. Prove that $f$ and $g$ are polynomials.

Let $z_{1}, \ldots , z_{n}$ be distinct complex numbers, and let $a_{1}, \ldots , a_{n}$ be nonzero complex numbers such that $S_{p} = \sum_{j=1}^{n} a_{j}z_{j}^{p} = 0$ for $p = 0, 1, \ldots , m-1$ but $S_{m} \neq 0$. Here $1 \leq m \leq n-1$. How many zeros does the rational function $f(z) = \sum_{j=1}^{n} \dfrac {a_{j}}{z-z_{j}}$ have in $\mathbb {C}$? Why is $m \geq n$ impossible.