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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 function $f$ on a compact interval, value truncation replaces large heights by a cap:One elementary convention requires every $f_r$ to be Riemann integrable and defines the truncated integral as ${\lim }_{r\to \infty }\int f_r$, allowing $+\infty$. The integrals are nondecreasing in $r$, so the limit exists in the extended real numbers; finiteness is an additional convergence condition. For nonnegative measurable functions, the monotone convergence theorem identifies this value with the Lebesgue integral whenever the capped Riemann integrals exist.

For $f(x)=x^{-1/2}$ on $(0,1]$, assign any finite value at zero. For $r\ge 1$, splitting at $x=r^{-2}$ givesThis removes a height cutoff. The usual improper integral instead removes a small endpoint interval, ${\lim }_{\epsilon \downarrow 0}{\int }_{\epsilon }^{1}f$; both conventions agree here. Neither convention justifies cancellation of two infinite positive and negative parts.

For a bounded real function $f$ and a partition of an interval $P=(a=x_0<\cdots <x_n=b)$, set $m_i={\inf }_{[x_{i-1},x_i]}f$ and $M_i={\sup }_{[x_{i-1},x_i]}f$. Its lower and upper Darboux sums areThey bracket every tagged Riemann sum on $P$. Partition refinement increases $L(f,P)$ and decreases $U(f,P)$, because smaller cells have larger infima and smaller suprema. A function is Riemann integrable exactly when these brackets can be made arbitrarily close.^[ref]

For $f(x)=x^2$ on $[0,1]$ with $x_i=i/n$, the monotonicity of $f$ gives $m_i=((i-1)/n{)}^2$ and $M_i=(i/n{)}^2$. Subtracting the sums yieldsThis computes an explicit approximation-error bracket without first knowing an antiderivative.

A bounded function is Riemann integrable exactly when for every $\epsilon >0$ some partition $P$ satisfies $U(f,P)-L(f,P)<\epsilon$.

For a bounded real function on a closed finite interval, the Riemann integral exists when the supremum of its lower Darboux sums equals the infimum of its upper Darboux sums. Their common value is ${\int }_a^{b}f(x)dx$. Equivalently, for every $ϵ>0$ there is a partition of an interval whose upper-minus-lower sum is less than $ϵ$.^[ref]

For $f(x)=x$ on $[0,1]$, uniform partitions into $n$ pieces give lower and upper sums $1/2-1/(2n)$ and $1/2+1/(2n)$, proving that the integral is $1/2$. The Dirichlet function has lower sums zero and upper sums one on this interval, so it is not Riemann integrable. The definition explains why boundedness alone is insufficient.

The total variation of a function $f:[a,b]\to \mathbb{R}$ measures its accumulated rises and falls, rather than just its change between the endpoints. For each partition of an interval $\Pi =(a=t_0<\cdots <t_n=b)$, add the absolute values of its sampled increments. Then take the supremum over all such partitions of an interval:The function has bounded variation when $V_a^b(f)<\infty$. This definition uses its value at every point of the interval, including both endpoints.^[ref]

For a nondecreasing function, all increments are nonnegative and the sum telescopes to $f(b)-f(a)$. More generally, the triangle inequality gives $|f(b)-f(a)|\le V_a^b(f)$. Equality can fail dramatically when the function changes direction: an object can travel a long distance and finish near its starting position. Here we measure movement of the function's value; arc length of its function graph also includes horizontal movement.

Every Riemann-integrable function on a compact interval can be approximated in $L^1$ by continuous functions. Consequently their integrals converge uniformly over all subintervals.

For a tagged partition $a=x_0<\cdots <x_n=b$ with $t_i\in [x_{i-1},x_i]$, the corresponding Riemann sum isIts limits over increasingly fine partitions define the Riemann integral.

In dimension greater than two, if sectional curvature is pointwise independent of the two-plane, the contracted Bianchi identity forces that curvature to be constant on each connected component.

The positive-definite specialization is Schur's lemma.

The Ricci tensor and scalar satisfy ${\nabla }^{\alpha }{R}_{\alpha \beta }=\frac{1}{2}{\nabla }_{\beta }R$.

The algebraic symmetries of the Riemann curvature tensor leave only one independent curvature component in two dimensions and forceConsequently the Einstein tensor vanishes identically.

A Drude response has a single relaxation pole, $G_R(\omega )={\sigma }_0/(1-i\omega {\tau }_0)$. Causality places the pole in the lower half plane when ${\tau }_0>0$.

The spectral function records the weighted energy differences connected by an operator. With the convention $G_R=-i\theta ⟨[A,A]⟩$, it is proportional to $-\mathrm{Im}{G}_R$.

In the circular restricted three-body problem the two massive bodies move on circular orbits about their barycentre. The problem is autonomous in their uniformly rotating frame.

The sperm number compares filament length with an elastohydrodynamic penetration length. For periodic forcing at angular frequency $\omega$, the usual convention isIts fourth power is a viscous-to-bending ratio on the full filament length. In steady cross-flow problems a different convention also occurs: $\mathrm{Sp}={\zeta }_{\perp }{L}^{3}U/A$, where $U$ is flow speed and $A$ the filament bending modulus. The defining formula distinguishes these conventions.

A stationary Gaussian process has a covariance positive-semidefinite kernel $k(x,y)=K(x-y)$. In the integrable Fourier transform setting on ${\mathbb{R}}^d$, use the conventionwith nonnegative integrable spectral density $\hat{K}$. Its Reproducing kernel Hilbert space has the weighted Fourier transform normThe admissible functions have finite displayed norm and $\hat{f}=0$ almost everywhere where $\hat{K}=0$. That support condition is essential: a zero spectral weight cannot supply a missing frequency by treating an undefined quotient as zero. Small positive spectral weights instead make corresponding components expensive.

The norm describes deterministic functions associated with the covariance positive-semidefinite kernel. It does not assert that a random sample path of the Gaussian process has finite norm in this space. Regularization and random-path variation therefore describe different aspects of the same covariance model.^[ref]