existence of roots for a general function over the complex plane

Question

For some more general functions other than polynomials, are there any fixed conditions for the existence of roots in a general sense?

For instance, function like $z\mathrm{sin}z-1$

Answer 1

If a function $f$ is holomorphic in $\Bbb C$ and $\lim_{z\to\infty} f(z)=\infty$, if $f$ has no roots then you can apply the Liouville's theorem to $1/f$. That is, if $\lim_{z\to\infty} f(z)=\infty$ and $f$ is entire, then $f$ has a root or is constant.

If $f$ is a non constant, entire function and skips some $w\in\Bbb C$ (that is, $f(z)\neq w$ for every $z\in \Bbb C$), then $w=0$ (which means that $f$ has no roots) or the Picard's theorem implies that $f$ has a root. In other words, if $f$ is a non constant, entire function and $f(z)-w$ has no roots for some $w\in\Bbb C\setminus\{0\}$, then $f(z)$ has a root.

I think (not sure) that your example doesn't meet any of these two conditions, but it certainly has roots, because the restriction $f(x)=x\sin x-1$ for $x\in\Bbb R$ is continuous and $f(0)=-1<0$, $f(\frac\pi2)=\frac\pi2-1>0$. In this case, the Bolzano's theorem implies that $f$ has root.