Distribution of the Zeros of a Function

By Rouché's theorem, the functions $\varphi (z)$ and $\psi (z)$, regular inside a contour, have the same number of zeros there if on the contour $\left|\varphi (z)-\psi (z)\right| < \left|\varphi (z)\right|$. Using this, find the number of roots lying in the disk $\left|z\right| < 1$ for the equation $z^{4}-5z+1 = 0$.

By Rouché's theorem, the functions $\varphi (z)$ and $\psi (z)$, regular inside a contour, have the same number of zeros there if on the contour $\left|\varphi (z)-\psi (z)\right| < \left|\varphi (z)\right|$. Using this, find the number of roots lying in the disk $\left|z\right| < 1$ for the equation $z^{6}-6z+10 = 0$.

By Rouché's theorem, the functions $\varphi (z)$ and $\psi (z)$, regular inside a contour, have the same number of zeros there if on the contour $\left|\varphi (z)-\psi (z)\right| < \left|\varphi (z)\right|$. Using this, find the number of roots lying in the disk $\left|z\right| < 1$ for the equation $z^{8}-4z^{5}+z^{2}-1 = 0$.

By Rouché's theorem, the functions $\varphi (z)$ and $\psi (z)$, regular inside a contour, have the same number of zeros there if on the contour $\left|\varphi (z)-\psi (z)\right| < \left|\varphi (z)\right|$. Using this, find the number of roots lying in the disk $\left|z\right| < 1$ for the equation $z = \varphi (z)$, where $\left|\varphi (z)\right| < 1$ for $\left|z\right| \leqslant 1$.

How many roots does the equation $z^{4}+2z^{3}+3z^{2}+z+2 = 0$ have in each square?

How many roots does the equation $2z^{4}-3z^{3}+3z^{2}-z+1 = 0$ have in each square?

Prove that for $\lambda > 1$ the equation $z = \lambda -e^{-z}$ has, in the half-plane $x > 0$, a unique root, and moreover a real one.

Prove that $z = ae^{z}$ has, for $0 < a < \frac{1}{e}$, only one root in the disk $\left|z\right| \leqslant 1$.

Prove that the equation $z^{2} = ae^{z}$, where $0 < a < \frac{1}{e}$, has two roots in the disk $\left|z\right| < 1$. They are real and of opposite signs.

Prove that the equation $1+z+az^{n} = 0$, where $n$ is an integer greater than one, has at least one root in the disk $\left|z\right| \leqslant 2$ for any $a$.

Prove that the equation $z = \tg z$ has only real roots: a triple root $z = 0$ and roots $z_{n} = (2n+1)\frac{\pi }{2}-\rho_{n}$, where $\rho_{n} > 0$, $\rho_{n} \to 0$ as $n \to \infty$.

Prove that the equation $\ch z\cos z = 1$ has a quadruple root $z = 0$ and roots of the form $\pm r_{n}$, $\pm ir_{n}$, $r_{n} = \left(n+\frac{1}{2}\right)\pi +(-1)^{n-1}\rho_{n}$, where $\rho_{n} > 0$ and $\rho_{n} \to 0$ as $n \to \infty$.

Prove that if the sequence of functions $\left\{ f_{n}(z)\right\}$, regular in a domain $G$, converges uniformly on compact subsets of this domain to some function $f(z) \not\equiv 0$, then every zero of the function $f(z)$ is the limit of corresponding zeros of the functions $f_{n}(z)$.

Prove that if, under the conditions of the previous problem (problem 5501), the function $f(z)$ has inside the domain $G$ a zero $z = z_{0}$ of multiplicity $m$, then one can find constants $\rho_{0} > 0$ and $C > 0$ such that, as soon as the relation $\left|f_{n}(z)-f(z)\right| < \rho < \rho_{0}$ holds on the boundary of the domain $G$, the function $f_{n}(z)$ has exactly $m$ zeros in the disk $\left|z-z_{0}\right| < C\sqrt[m]{\rho }$.

Prove that if $\mathrm{Im}\left(z_{\nu }\right) > 0$, and $\prod_{\nu =1}^{n}(z-z_{\nu }) = \varphi (z)+i\psi (z)$, where $\varphi (z)$ and $\psi (z)$ are polynomials with real coefficients, then the roots of these polynomials are real, simple, and interlacing.

Prove that under the same conditions (problem 5503) and for $a > 0$, the roots of the equations

have the same properties.

Prove that the number of roots of the real trigonometric polynomial $\sum_{\nu =m}^{n}(a_{\nu }\cos \nu \varphi +b_{\nu }\sin \nu \varphi )$, in the interval $0 \leqslant \varphi < 2\pi$, is not less than $2m$ and not more than $2n$.

Prove that for $a_{0} < a_{1} < a_{2} < \cdots < a_{n}$ it is easy to prove, by multiplying by $1-z$, that the polynomial $\sum_{\nu =0}^{n} a_{\nu }z^{\nu }$ has no roots in the region $\left|z\right| \geqslant 1$. Using this, show that the trigonometric polynomial $\sum_{\nu =0}^{n} a_{\nu }\cos \nu \varphi$, with the same $a_{\nu }$, has exactly $2n$ distinct roots in the interval $0 \leqslant \varphi < 2\pi$ (as does the polynomial $\sum_{\nu =1}^{n} a_{\nu }\sin \nu \varphi$).

Let, for $0 < t < a$, the function $f(t)$ be positive and nondecreasing, and let the integral $\int_{0}^{a} f(t)\, dt$ make sense. Prove that the entire function $f(z) = \int_{0}^{a} f(t)\cos zt\, dt$ has only real zeros. (Pólya's Theorem.)

Prove that under the same conditions (problem 5507) and for $b > 0$, the entire functions

also have only real zeros.

Let $f(z)$ be regular in the disk $\left|z\right| \leqslant r$ and have, inside the disk, roots $a_{1}, a_{2}, \ldots , a_{n}$, distinct from zero. By joining each of the zeros $a_{\nu }$ with a cut to the circle $\left|z\right| = r$ and integrating the corresponding function over the resulting contour, prove Jensen's formula:

Prove that if, under the conditions of the previous problem (problem 5509), the maximum of $\left|f(z)\right|$ equals $M$, $\left|f(0)\right| = m$, and $N$ is the number of zeros of $f(z)$ in the disk $\left|z\right| \leqslant \frac{r}{e}$, then

Prove the theorem: the roots of the function

where $a > b > \cdots > m$, and the numbers $A$ and $M$ are nonzero, lie in the strip $-\sigma < \mathrm{Re}\left(z\right) < \sigma$, and $N$, the number of them in the rectangle $-\sigma < \mathrm{Re}\left(z\right) < \sigma$, $0 < \mathrm{Im}\left(z\right) < T$, is expressed by the formula

where $O(1)$ is bounded as $T \to \infty$.