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The completed zeta functionis an entire function satisfying $\xi (s)=\xi (1-s)$. Thus every nontrivial zero $\rho$ is accompanied by $1-\rho$, $\overline{\rho }$, and $1-\overline{\rho }$.

A nodal crossing locally consists of two complex branches meeting transversely and is not a one-dimensional complex manifold at the intersection.

For a plane algebraic curve this crossing is an ordinary double point, with two distinct tangent directions.

A homeomorphism to a complex manifold transports its atlas and thereby defines a complex structure on the source.

Removing a closed discrete set from a Riemann surface leaves an open complex one-manifold; if connected, it is again a Riemann surface.

The Riemann sphere is $\mathbb{C}\cup \{\infty \}$ with two charts related by reciprocal coordinates.

The extra point completes the complex plane into a compact Riemann surface. It is one point shared by every direction of escape: $n$, $-n$ and $in$ all tend to $\infty$ as $n\to \infty$. A basis of neighbourhoods at $\infty$ isThis is the one-point compactification of the complex plane; convergence to $\infty$ means that the modulus tends to infinity.^[ref]

For a concrete picture, use the unit sphereand identify $z=x+iy$ with $(x,y,0)$ in its equatorial plane. The north pole $N=(0,0,1)$ represents $\infty$ and the south pole $S=(0,0,-1)$ represents $0$. The stereographic projection below gives the identification. This convention fixes the scale: the equator corresponds to $|z|=1$.^[ref]

Why add this point? The Möbius transformation $z\mapsto 1/z$ then exchanges $0$ and $\infty$ and becomes a holomorphic map everywhere on the Riemann sphere. Its apparent blow-up at zero disappears when its output is expressed in the reciprocal chart at infinity: $w=1/(1/z)=z$. This is a concrete instance of a meromorphic function as a holomorphic map to the Riemann sphere. The additional point makes the complex plane compact, while its second manifold chart lets us do ordinary complex analysis near infinity.

For a degree-$d$ holomorphic map, $2g_X-2=d(2g_Y-2)+\sum (e_p-1)$.

Let $f:X\to Y$ be a nonconstant holomorphic map between compact Riemann surfaces that are connected spaces, without boundary. Write $g_X,g_Y$ for their genera and $d\ge 1$ for the degree of a holomorphic map. With analytic ramification index $e_p$ at each $p\in X$, the formula isHere $\chi =2-2g$ is the Euler characteristic. Only finitely many ramification points contribute. The holomorphic map is automatically surjective; the valency theorem gives ${\sum }_{p\in f^{-1}(y)}{e}_p=d$ even at a branch value of a holomorphic map. Constant maps are excluded.^[ref]

The useful interpretation is that total ramification of a holomorphic map measures how much the source Euler characteristic falls short of $d$ copies of the target. The ramification of a power map of the Riemann sphere spends that entire correction at two points; the double cover of the sphere with four branch values spends it at four points and produces a genus-one surface.

For a nonconstant analytic map $f:R\to S$ between compact connected Riemann surfaces, there is an integer $\mathrm{deg}f$ such that for every $w\in S$,

The degree counts sheets with multiplicity of the associated branched covering.

A complete analytic function is the collection of all continuations of a germ. Its Riemann surface consists of these germs, with projection to their base points.

A covering is regular when its deck group acts transitively on each fibre, equivalently when its fundamental-group subgroup is normal.

A real-valued function on a Riemann surface is harmonic when its expression in every holomorphic coordinate chart has vanishing planar Laplacian.

Two holomorphic functions on a connected Riemann surface that agree on a set with an accumulation point agree everywhere. Local charts reduce the proof to isolated zeros of a one-variable holomorphic function.

Every simply connected Riemann surface is biholomorphic to the Riemann sphere, the complex plane, or the unit disc.

If $\pi :S\to R$ is a covering of a Riemann surface, compose every chart on an evenly covered open set with each local inverse sheet of $\pi$. The resulting transition maps are those of $R$, so they define a unique complex structure making $\pi$ a local biholomorphism.

A germ at $z\in D$ is an equivalence class $[f{]}_z$ of holomorphic functions defined near $z$, where two representatives are equivalent when they agree on some neighbourhood of $z$.

It is the holomorphic-function case of a germ.

A map between Riemann surfaces is holomorphic when its expression in every pair of local complex coordinates is a holomorphic function.

For $2\le p<\infty$,Thus every function in $L^2\cap L^{\infty }$ belongs to every $L^p$ between them.

If a radial function on ${\mathbb{R}}^n$ behaves like $r^a$ near zero and like $r^{-b}$ at infinity, then its $q$th power is locally integrable at zero exactly when $aq>-n$ and integrable at infinity exactly when $bq>n$. This follows from the radial measure factor $r^{n-1}dr$.

A smooth compressive simple wave forms a shock when characteristics first intersect. If characteristics are parametrized by their emission time $\tau$, the first shock occurs at the minimum positive time for which $\partial x(\tau ,t)/\partial \tau =0$.

In a right-moving simple wave entering gas initially at rest with sound speed $c_0$, the other Riemann invariant is constant, soThe isentropic relationconverts the velocity disturbance into its nonlinear pressure disturbance.

For a nonnegative unbounded function, one truncation convention sets $f_r=\min (f,r)$ and asks that every $f_r$ be Riemann integrable and that $\int f_r$ have a finite limit as $r\to \infty$.