For integer $m\ge 0$, $J_{-m}(z)=(-1{)}^m{J}_m(z)$. Thus negative integer orders remain regular at zero. The familiar $J_m(z)\sim z^m/(2^{m}m!)$ formula is used for nonnegative order; a negative factorial is not a valid continuation.

For the Bessel differential equation $x^2{y}^{\prime \prime }+x{y}^{\prime }+(x^2-{\nu }^2)y=0$, substitute $y=x^s{\sum }_{n\ge 0}{a}_n{x}^n$. The indicial equation is $s^2-{\nu }^2=0$, and coefficients obey $[(n+s{)}^2-{\nu }^2]a_n=-a_{n-2}$. For $s=\nu \ge 0$, $a_1=0$ and all odd coefficients vanish. Iteration gives $a_{2m}=(-1{)}^m{a}_0\Gamma (\nu +1)/[4^{m}m!\Gamma (m+\nu +1)]$. Choosing $a_0=1/[2^{\nu }\Gamma (\nu +1)]$ gives the Bessel function of the first kind $J_{\nu }$.

For noninteger $\nu$, choosing the even-power series with $s=-\nu$ replaces $m+\nu$ by $m-\nu$ and yields $J_{-\nu }$ with its standard normalization. These solutions are independent. For integer $\nu \ge 0$, the first solution has coefficients $(-1{)}^m/[2^{2m+\nu }m!(m+\nu )!]$ and defines an entire function: the term ratio tends to zero at every finite complex argument. For noninteger order, the even analytic series converges everywhere, while the prefactor $x^{\pm \nu }$ requires a branch off the positive real axis.

If the integrand $L(q_1,\dots ,q_m,q_1^{\prime },\dots ,q_m^{\prime })$ has no explicit dependence on its independent variable $z$, its Euler-Lagrange equations imply the displayed first integral. Indeed, differentiate the left side and cancel the second-derivative terms to obtain ${\sum }_j{q}_j^{\prime }({\partial }_{q_j}L-d({\partial }_{q_j^{\prime }}L)/dz)=0$. This is the several-variable form of the Beltrami identity. Applied to Fermat principle, it expresses conservation of the component of the optical momentum of a ray along a translation-invariant direction.

A two-setting nonlocality argument in which several zero-probability events forbid a further coincidence in any deterministic local hidden-variable model, but quantum theory assigns that coincidence positive probability. For binary indicators, suppose $C_{+}^R{C}_{-}^R=0$, $D_{+}^I=1\Rightarrow C_{-}^R=1$ and $D_{-}^I=1\Rightarrow C_{+}^R=1$. Any assignment with $D_{+}^I{D}_{-}^I=1$ would violate the first relation. A quantum implementation can nevertheless give such a coincidence. Real imperfect experiments require an inequality that accounts for their nonzero error probabilities. Lucien Hardy gave a two-particle inequality-free argument in 1992.^[ref]

Quantum correlations incompatible with Bell local causality under additional assumptions including measurement independence. This is an operational statement about joint probabilities, not controllable faster-than-light signaling: quantum predictions also obey quantum no-signalling. Entanglement alone does not guarantee violation of a specified Bell inequality in a given measurement arrangement.

Costs of acquiring information, maintaining responsiveness or switching action. They must be distinguished from a maladaptive action caused by an unreliable cue.

The gap feature in a superconducting single-particle excitation spectrum. In a simple BCS model, excitations have $E_{\mathbf{k}}=\sqrt{{\xi }_{\mathbf{k}}^2+|\Delta (\mathbf{k}){|}^2}$. The superconducting gap function specifies the pairing amplitude and its phase; its magnitude need not be constant, and nodes can leave gapless excitations.

The isotropic weak-coupling BCS theory predicts the gap ratio approximately $3.528$ and the electronic heat-capacity jump approximately $1.426$ in units of the normal-state heat capacity at the transition. These compare distinct measurements of the same paired state. They are conditional predictions of the weak-coupling fully gapped model, rather than universal numbers for all superconductors.

An attractive local pairing interaction can be decoupled with a complex Hubbard–Stratonovich transformation. Integrating the fermionic Grassmann variables gives an effective action for the pair field. Its uniform saddle is the BCS gap equation, and its slow-field expansion near the transition gives the Ginzburg-Landau theory of superconductivity. A trace over a fully doubled particle-hole basis requires the corresponding factor of one half; the displayed formula uses one independent pairing block.

For a constant effective density of states and attractive coupling $\lambda$, let the linearized BCS theory gap equation integrate $\tanh (\xi /(2k_B{T}_c))/(2\xi )$ over $-\sigma <\xi <\theta$, with positive cutoffs. Its even integrand makes the result the mean of two positive-cutoff logarithms. When $k_B{T}_c\ll \sigma ,\theta$, $1/\lambda \simeq \frac{1}{2}\ln (1.13\theta /(k_B{T}_c))+\frac{1}{2}\ln (1.13\sigma /(k_B{T}_c))$, giving the displayed geometric mean of cutoffs.

If a band bottom truncates the lower side, $\sigma =\min (\mu ,\theta )$ for positive chemical potential $\mu$. The weak-coupling limit at fixed $\mu /\theta >0$ must not be interchanged with an uncontrolled $\mu \to 0$ extrapolation: the lower logarithm requires $\mu \gg k_B{T}_c$ when $\mu <\theta$.

The continuous evolution, in suitable attractive Fermi systems, from weakly paired overlapping Cooper pairs to tightly bound composite bosons undergoing Bose-Einstein condensation. Pair formation, phase coherence and the chemical potential must be determined consistently across the crossover. A weak-coupling gap-temperature formula with fixed density of states is not uniformly valid through the molecular limit.

A fermionic excitation obtained by a canonical particle–hole Bogoliubov transformation of a paired mean-field Hamiltonian. The uniform spin-singlet problem has the displayed positive energy and weights $|u{|}^2=(1+\xi /E)/2$, $|v{|}^2=(1-\xi /E)/2$. It interpolates between particle-like and hole-like excitations across the Fermi surface. These single-particle excitations differ from collective phase and amplitude modes of the order parameter.

If inclusion depends only on fully observed data $d$, then $p(S\mid d,\theta )=p(S\mid d)$ cancels from Bayes' theorem for that individual's parameters. Selection can still matter for unknown population parameters and for missing or unconditioned features. For independent detected observations conditional on their count and population parameters $\Lambda$, each likelihood has normalization $\alpha (\Lambda )=\int S(d)p(d\mid \Lambda )dd$. A model for the detected count adds its own count likelihood. Do not combine a selected-population prior with a truncated-data likelihood in a way that counts the same selection twice.

nucleic acid complementarity is the relation between sequences that permits specified base pairs to form when aligned with appropriate strand orientation. In homologous recombination, structural organization helps test pairing rather than reading one fixed base sequence.

Watson-Crick base pairing is the canonical geometry of pairing between complementary nucleic-acid bases, including adenine–thymine and guanine–cytosine in ordinary DNA. It is a particular pairing geometry, not a synonym for every possible base pair.