Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{Z}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ALL zeroes of the corresponding $f(z)$ belong to coordinate axes in $z$. Formally

$$ D = \lbrace (a,b) \, \small| \exists z\in \mathbb{C} \, f(z) =0 \, \wedge \, {\rm arg} (z) \neq 0,\pi/2,\pi,3\pi/2 \rbrace. $$

Obviously the line $b=0$ is not in $D$, as zeroes of $\cos(z)\cosh(az)=0$ belong to axes $Re z=0, Im z=0$.

Problem: define geometrically the set $D$.

Calculations show that a set $D$ is finite near the origin. How to describe it more exactly? Is it possible to find a ball centred at origin for which $D$ is inside this ball? Is it possible to find estimates for a radius of this ball? More suggestions?

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{C}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ALL zeroes of the corresponding $f(z)$ belong to coordinate axes in $z$. Formally

$$ D = \lbrace (a,b) \, \small| \exists z\in \mathbb{C} \, f(z) =0 \, \wedge \, {\rm arg} (z) \neq 0,\pi/2,\pi,3\pi/2 \rbrace. $$

Obviously the line $b=0$ is not in $D$, as zeroes of $\cos(z)\cosh(az)=0$ belong to axes $Re z=0, Im z=0$.

Problem: define geometrically the set $D$.

Calculations show that a set $D$ is finite near the origin. How to describe it more exactly? Is it possible to find a ball centred at origin for which $D$ is inside this ball? Is it possible to find estimates for a radius of this ball? More suggestions?

I knew of this problem some years ago from Prof. A.P.Soldatov. This problem occurs as solvability condition of some problems for Lavrent'ev-Bitsadze mixed type differential equation.

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{Z}, a,b \in \mathbb{R}$. Denote D being a set {a,b} of such pairs of parameters that NOT ALL zeroes of corresponding $f(z)$ belong to coordinate axes in $z$. Obviously the line $b=0$ are not in D, as zeroes of $\cos(z)\cosh(az)=0$ belong to axes $Re z=0, Im z=0$.

Problem: define geometrically the set D in the plane {a,b} as exact as possible.

Calculations show that a set D is finite near the origin. How to describe it more exact? Is it possible to find a ball centred at origin for which D is inside this ball? Is it possible to find estimates for a radius of this ball? More suggestions?

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{Z}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ALL zeroes of the corresponding $f(z)$ belong to coordinate axes in $z$. Formally

$$ D = \lbrace (a,b) \, \small| \exists z\in \mathbb{C} \, f(z) =0 \, \wedge \, {\rm arg} (z) \neq 0,\pi/2,\pi,3\pi/2 \rbrace. $$

Obviously the line $b=0$ is not in $D$, as zeroes of $\cos(z)\cosh(az)=0$ belong to axes $Re z=0, Im z=0$.

Problem: define geometrically the set $D$.

Calculations show that a set $D$ is finite near the origin. How to describe it more exactly? Is it possible to find a ball centred at origin for which $D$ is inside this ball? Is it possible to find estimates for a radius of this ball? More suggestions?