The equilibrium equations are $x(1+x-2y)=0$ and $(x-1)y=0$. If $x=0$, the second equation gives $y=0$. Otherwise either $y=0$, giving $x=-1$, or $x=1$, giving $y=1$. Thus there are exactly three equilibria. The Jacobian isAt $(0,0)$ it is $\mathrm{diag}(1,-1)$: a saddle. At $(-1,0)$ it is $\left(\begin{array}{cc}-1& 2\\ 0& -2\end{array}\right)$, with eigenvalues $-1,-2$: a stable node. At $(1,1)$ it is $\left(\begin{array}{cc}1& -2\\ 1& 0\end{array}\right)$, with characteristic polynomial ${\lambda }^2-\lambda +2$ and eigenvalues $(1\pm i\sqrt{7})/2$: an unstable spiral. All three are hyperbolic, so the nonlinear local classifications follow from linearization.

A critical point is an equilibrium: $\boxed{f(a,b)=g(a,b)=0}$. Put $X=x-a$, $Y=y-b$. The Jacobian matrix $J$ gives the first-order linearized equation $(\dot{X},\dot{Y}{)}^{\mathsf{T}}=J(X,Y{)}^{\mathsf{T}}$. The following eigenvalue tests classify that linearized system. For a continuously differentiable nonlinear system, the cases with nonzero real parts persist as the usual local node, saddle or spiral classifications. The purely imaginary case needs extra nonlinear information, as explained in (iv).

The domain has two connected components. The zeros force the function to vanish on the upper half-plane but impose nothing on the lower one. DefineThis function is locally constant and hence holomorphic on the stated open set, and is not identically zero on that whole set. Thus the answer is yes. Connectedness is exactly the missing hypothesis for a global identity conclusion.

Here the zeros accumulate only at the boundary point zero. An explicit holomorphic, nonzero example on the upper half-plane isAt $z_n=i/(n+1)$ its argument is $\pi (n+1)$, so it vanishes. It is holomorphic because the only excluded point of its formula is zero; it is not identically zero, for instance at $z=2i$. Thus the answer is yes.

The given distinct zeros converge to $i$, which lies inside the connected upper half-plane. The identity theorem for holomorphic functions forces the function to vanish identically. Thus $\boxed{\text{no}}$.

The denominator vanishes at $z_k=1/(2\pi ik)$ for every nonzero integer $k$. These points approach zero, the denominator has a simple zero at each, and $\sin z_k\ne 0$ because a nonzero purely imaginary number is not a zero of sine. Hence there are poles arbitrarily close to zero. The origin is a nonisolated singularity, so it has no residue defined by a Laurent series on a punctured disc. In particular it must not be classified as an isolated essential singularity merely because $e^{1/z}$ has one.

Use $\sin z=z+O(z^3)$ andTheir product is $z^{-2}+(3/2)z^{-1}+O(1)$. Thus the singularity is a pole of order two, with $\boxed{{\mathrm{Res}}_{0}f=3/2}$.

Both numerator and denominator have simple zeros at zero, and $1-e^{-z}=z+O(z^2)$. Their quotient tends to one and extends holomorphically with that value. Hence the singularity is removable, and $\boxed{{\mathrm{Res}}_{0}f=0}$ because its Laurent series has no negative powers.

If a holomorphic bijection existed, its inverse would be an entire function taking values in the unit disc. Liouville theorem would make that inverse constant, contradicting bijectivity. Thus no holomorphic bijection exists.

A continuous bijection does exist, even a homeomorphism:Both maps preserve the argument and change the radius by the mutually inverse strictly increasing functions $r\mapsto r/(1-r)$ and $s\mapsto s/(1+s)$. The displayed formulas also prove continuity at zero and everywhere else.

Let $h=f^{-1}$, $w_0=f(z_0)$ and $z=h(w)$. Continuity of $h$ gives $z\to z_0$ as $w\to w_0$, and injectivity gives $z\ne z_0$ when $w\ne w_0$. ThereforeThe assumed nonvanishing derivative makes this limit well defined. It exists at every $w_0\in V$, so $h$ is holomorphic, with $\boxed{h^{\prime }(w)=1/f^{\prime }(h(w))}$.

Squaring maps the first quadrant bijectively onto the upper half-plane. For $\zeta$ in that half-plane, $|\zeta -i|<|\zeta +i|$, and the Cayley transform $(\zeta -i)/(\zeta +i)$ maps it bijectively onto the unit disc. Their composition isThe inverse is the unique first-quadrant square root of $i(1+w)/(1-w)$, and is holomorphic by the first argument. Hence $H(w)=g(F^{-1}(w))$ is holomorphic throughout the disc. Its Taylor series at zero converges for $|w|<1$, locally uniformly there. Substituting $w=F(z)$ gives the stated expansion, with $c_n=H^{(n)}(0)/n!$, convergent at every point of the quadrant and locally uniformly on it.

A Euclidean domain is an integral domain equipped with a function $\delta :R\setminus \{0\}\to \mathbb{N}$ such that for every $a\in R$ and nonzero $b\in R$ there are $q,r\in R$ with $a=qb+r$ and either $r=0$ or $\delta (r)<\delta (b)$. One may additionally require $\delta (a)\le \delta (ab)$ for nonzero $a,b$; degree on a polynomial ring also satisfies this convention.

An integral domain is a commutative ring with identity $1\ne 0$ and no zero divisors: $ab=0$ implies $a=0$ or $b=0$. Equivalently nonzero elements admit cancellation in multiplication.

For equal-length crystallographic root configurations in a plane, taking $u=v$ in the closure condition gives $S_v(v)=-v$, so every vector occurs with its negative. For fixed $v\in \Phi$, a vector $u\in \Phi$ has $⟨u,v⟩\in \{-1,-1/2,0,1/2,1\}$, because both vectors are unit vectors. Thus its angle relative to $v$ belongs to the finite list $0,\pm \pi /3,\pm \pi /2,\pm 2\pi /3,\pi$.

If all vectors are collinear, the set is exactly $\{v,-v\}$, giving two opposite points of the unit circle.

If an orthogonal vector $w$ occurs, closure supplies $\pm v,\pm w$. A unit vector making an angle of $\pi /3$ or $2\pi /3$ with $v$ would have inner product $\pm \sqrt{3}/2$ with $w$, contradicting integrality of twice that inner product. Thus these four points are the whole set: the vertices of a square.

In the remaining case a noncollinear pair makes an angle of $\pi /3$ or $2\pi /3$. Changing a sign, take their angles as $0,\pi /3$. Reflecting the second in the line perpendicular to the first supplies the angle $2\pi /3$, and negation supplies the other three hexagon vertices. No perpendicular vector can occur, by the preceding inner-product obstruction. The initial finite list now leaves only these six points. Conversely the opposite pair, square and regular hexagon each satisfy both reflection closure and the integrality condition. Hence

Write $n=\dim V$, which is finite by hypothesis. This supplies the meaning of the $n$ used later in the variant; the proof is otherwise unchanged.

The functional $\ell (u)=⟨u,v⟩$ is linear and nonzero because $\ell (v)=⟨v,v⟩>0$. It therefore has rank one, and rank-nullity theorem givesNo Gram–Schmidt argument is needed. Every vector decomposes uniquely as $u=w+av$, where $w\perp v$ and $a=⟨u,v⟩/⟨v,v⟩$. Substitution gives $S_v(w+av)=w-av$. Thus the eigenspace for $-1$ is $\mathbb{R}v$, and that for $+1$ is $v^{\perp }$. The latter is an actual eigenvalue when $n\ge 2$; when $n=1$ its eigenspace is zero and only $-1$ occurs. In the plane the map is reflection in the line perpendicular to $v$.