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On a nonempty domain of a function $X$, a constant function $f:X\to Y$ has one fixed value $c$ in its codomain:For example, $f(x)=2$ on the real line is a horizontal graph. Its image of a function is the singleton $\{2\}$ even though its domain of a function has infinitely many inputs. A constant function from the real line to the real numbers has derivative zero and every nonzero real period of a function; there is no smallest positive period. It supplies a reference level from which variation can be measured.

A constant map between vector spaces is a linear map only when its value is the zero vector, since a linear map must preserve zero. A nonzero constant map is still an affine map. In a Fourier series, the constant term records the period average, while the other terms describe oscillation around that level.

A real- or complex-valued function $f:X\to \mathbb{C}$ is bounded when one finite constant controls all its values:For example, $\sin x$ is bounded by one on the real line. The reciprocal $1/x$ is bounded by one on $[1,\infty )$, but is an unbounded function on $(0,1)$, showing that the domain of a function matters. Each value at an allowed input may be finite even when no common bound exists.

A piecewise linear function agrees with an affine function on each member of a finite or locally finite family of subsets partitioning its domain of a function. On the real line the pieces are intervals; in higher dimensions each piece can be a polyhedron. Here the conventional word “linear” permits a constant offset within each piece.

For example, the absolute value function satisfies $|x|=-x$ for $x\le 0$ and $|x|=x$ for $x\ge 0$. Both branches agree at zero, so the resulting function is continuous, although its derivative changes there. Joining sampled values by linear interpolation gives another common construction. If the partition pieces meet, their formulas must agree on overlaps to define a single function; allowing separate pieces across a jump is a convention that should be stated.

The identity function on a set $X$ maps each $x\in X$ to itself: ${\mathrm{id}}_X(x)=x$. It changes no information. For every function $f:X\to Y$, function composition satisfies $f\circ {\mathrm{id}}_X=f$ and ${\mathrm{id}}_Y\circ f=f$. The subscripts specify the appropriate source or target set. An inverse function reverses a bijection exactly when the two possible compositions are these identity functions.

When $1/p+1/q+1/r<1$, the orientation-preserving hyperbolic triangle group has presentationand is a non-elementary Fuchsian group.

This is the orientation-preserving hyperbolic member of the broader triangle group construction.

A nonidentity orientation-preserving isometry of the hyperbolic plane is parabolic when it has one fixed point on the ideal boundary and none in the plane. In the upper half-plane model, every parabolic isometry is conjugate to a nonzero horizontal translation.

This is one type of isometry of the hyperbolic plane, classified by its fixed points.

The Gauss hypergeometric equation isIts only possible singular points on the Riemann sphere are $0,1,\infty$, and they are regular singular points when singular. Their characteristic exponent pairs are $(0,1-c)$, $(0,c-a-b)$ and $(a,b)$ respectively, with exponents at infinity defined by behavior $z^{-\rho }$. The sum is one, agreeing with the Fuchs relation.^[ref]

When $c\notin \{0,-1,-2,\dots \}$, the solution analytic at zero with value one is the Gauss hypergeometric function ${}_2{F}_1(a,b;c;z)$. Its coefficient recurrence comes directly from matching powers. Resonant parameter values require compatibility checks or logarithmic companions; the normalized exponent-zero series cannot simply be asserted for every $c$.

For $a=0$ the constant function one is a solution. For a nonpositive-integer numerator parameter, the normalized series can terminate to a polynomial. The equation provides one common framework for many elementary and special functions.

A Papperitz or P-symbol lists the three regular singular points of a second-order equation and the two characteristic exponents at each.

This symbol records the local exponents of Riemann's differential equation.

A singular point of a linear differential equation is irregular when it fails the regular-singular criterion. For $y^{\prime \prime }+P(x)y^{\prime }+Q(x)y=0$, a pole of order greater than one in $P$ or greater than two in $Q$ makes the point irregular.

For $y^{\prime \prime }+P(z)y^{\prime }+Q(z)y=0$, assume $z_0$ is a singular point, rather than an ordinary point. It is a regular singular point exactly whenextend holomorphically across $z_0$. If $P,Q$ themselves extend holomorphically, the point is ordinary and should not be classified as singular merely because the displayed products are also analytic.

Writing $p_0=p(z_0)$ and $q_0=q(z_0)$, substitution of $y=(z-z_0{)}^r$ into the lowest-order terms gives the indicial equation $r(r-1)+p_{0}r+q_0=0$. A Frobenius solution then accounts for the analytic higher-order coefficients.

For $y^{\prime \prime }+a{y}^{\prime }/z+by/z^2=0$ at zero, the equation is regular singular unless both coefficients vanish, with indicial polynomial $r(r-1)+ar+b$. A coefficient such as $P(z)=z^{-2}$ instead violates the criterion and gives an irregular singular point.

Substitution of a Frobenius behavior $(z-z_0{)}^{\rho }$ into the leading singular terms gives the indicial equation and its characteristic exponents.

For $y^{\prime \prime }+P(z)y^{\prime }+Q(z)y=0$, the finite point $z_0$ is ordinary when $P$ and $Q$ are holomorphic at $z_0$.

In a flat radiation-plus-$\Lambda$ universe,The expansion changes from deceleration to acceleration when the radiation and cosmological-constant terms are equal, at

With zero cosmological constant, the condition $\rho +3P\ge 0$ implies $\ddot{a}\le 0$ andwherever $H\ne 0$.

For positive spatial curvature, zero cosmological constant, and $\rho ,P\ge 0$, continuity makes $\rho a^3$ nonincreasing during expansion. The negative curvature term then prevents unbounded growth of $a$, while the acceleration equation forces $\dot{a}$ to reach zero in finite time.

For a homogeneous pressureless fluid in comoving coordinates, the Euler equation giveswhile the Poisson equation givesTaking the divergence of the first relation and comparing them yieldsThis local fluid derivation describes a homogeneous infinite universe without choosing a physical centre of expansion.

Let $\alpha (s)$ be a regular curve parametrized by arc length on the unit sphere, with Frenet frame $(T,N,B)$ and convention $B^{\prime }=-\tau N$. Differentiating $⟨\alpha ,\alpha ⟩=1$ gives $⟨\alpha ,T⟩=0$ and $⟨\alpha ,N⟩=-1/\kappa$. ThusDifferentiating the second inner product identity using the Frenet-Serret formulas gives $\tau \beta ={\kappa }^{\prime }/{\kappa }^2$. Wherever $\tau \ne 0$,For the outward surface unit normal vector $\alpha$, the signed geodesic curvature isBoth plus signs follow from the stated torsion of a curve and surface orientation conventions. Also $1={\kappa }^{-2}+{\beta }^2$, hence ${\kappa }^2=1+{\kappa }_g^2$. The identities involving $\beta$ remain valid when $\tau =0$; the divided formulas need not. A non-equatorial latitude is a planar circle with zero torsion of a curve but nonzero geodesic curvature.

A scalar assignment $Q(\gamma ,s)$ is a pointwise Euclidean invariant in the weak covariance sense when it is unchanged by proper Euclidean motions and satisfiesunder translation of the arc-length parameter.

The unit-speed circular helixhas constant curvature $a/(a^2+b^2)$ and constant torsion $b/(a^2+b^2)$.

A Euclidean motion has the form $E(x)=Ax+b$ with $A\in O(3)$. It preserves Euclidean distance, angles, and all extrinsic geometric quantities transformed with their orientations.

This is the dimension-three case $E(3)=O(3)⋉{\mathbb{R}}^3$ of the Euclidean group.