Functions

In General > s.a. analysis [continuity classes, examples]; Approximation Methods; functional analysis [function spaces].
* Idea: A map f : M → $$\mathbb K$$ from a manifold M to a field $$\mathbb K$$ (usually, $$\mathbb K$$ = $$\mathbb R$$ or $$\mathbb C$$, and M may be $$\mathbb R$$ or $$\mathbb C$$ too).
@ General references: Oldham et al 08 [An Atlas of Functions]; Olver et al 10 [NIST handbook].
@ Random functions: Shale JFA(79) [of Poisson type]; Wang & Battefeld a1607 [generation, Dyson Brownian Motion algorithm].

Polynomials > s.a. Algebraic Geometry [decomposition of polynomials]; graph and knot invariants.
* Lee-Yang circle theorem: A somewhat mysterious result on the location of zeros of certain polynomials in statistical mechanics.
* Applications: Knot theory; Graph counting; Statistical mechanics.
* Operations: Notice that polynomial multiplication is a form of convolution.
* Monic polynomial: A univariate polynomial in which the leading coefficient is equal to 1.
* Grace-like polynomial: A polynomial P(z1, ..., zm, w1,..., wn), separately of degree 1 in each of its m + n arguments, such that P(z1, ..., wn) ≠ 0 whenever there is a circle in $$\mathbb C$$ separating z1, ..., zm from w1, ..., wn.
@ General references: Landau NAMS(87) [factoring]; Milovanović et al 94 [extremal problems, inequalities, zeros]; Ruelle mp/00 [grace-like]; Wang & Yeh JCTA(05) [with real zeroes].
@ Lee-Yang circle theorem: Ruelle PRL(71) [extension].
@ Random: Forrester & Honner JPA(99) [statistics of zeros]; Zelditch mp/00-proc.
@ Other types: Edwards BAMS(09) [solvable]; > s.a. Chebyshev, Hermite, Jack, Laguerre, legendre, Macdonald Polynomials.
> Polynomial approximations: see bessel functions.

Other Types > s.a. analytic functions; Almost Periodic Functions; Meromorphic Functions; Quasiperiodic Functions; series; summations.
* Bounded variation: A function f on an interval [a, b] is of bounded variation if there is a number M such that, for every partition a < x1 < x2 < ... < xn–1 < b of the interval, the sum of the variations |f(xi) – f(xi–1)| over all subintervals does not exceed M.
* Concave: A function f : [a, b] → $$\mathbb R$$ is concave iff the segment joining any two points in its graph lies below the curve, or for all x, y ∈ [a, b] and all λ ∈ [0,1], the values of f inside the interval satisfy f(λx + (1–λ) y) ≤ λ f(x) + (1–λ) f(y) .
* Convex: A function f : [a, b] → $$\mathbb R$$ is convex iff the segment joining any two points in its graph lies above the curve, or for all x, y ∈ [a, b] and all λ ∈ [0,1], the values of f inside the interval satisfy f(λx + (1–λ) y) ≥ λ f(x) + (1–λ) f(y) .
* Positive pure frequency function: One of the form F(x, t) = f(x) exp{–iωt}, or satisfying $$\cal L$$t f = –iω f, for ω > 0.
* Positive frequency function: One that can be extended to an analytic function in the lower complex t-plane.
> Other special types: see harmonic functions; L-functions; Rational Functions; Superoscillating Functions.
> Special functions: see Airy; bessel; Elliptic; Gamma; Hypergeometric; Jost; Mathieu; Struve; Theta; Whittaker; Zeta Function; spherical harmonics.

Examples and Properties > s.a. Germ of a Function; Hyperbolic Functions; trigonometry.
* Other examples: C function of compact support,

f(x):= exp{–(xx0)2/[(xx0)2h2]} ,   ψ(x):= exp{–1/x2(ax)2}   for x ∈ (0, a),   0 otherwise;

C function vanishing for x ≤ 0 and equal to 1 for xa,

χ(x):= C–1 –∞x ψ(x') dx' ,   C:= $$\,_{\mathbb R}^~$$ψ(x) dx ;

C function on $$\mathbb R$$n, of compact support, equal to 1 in a square box xi ∈ (αi, βi),

g(x1, ..., xn):= ∏i=1n χ(xiαi+a) χ(βixi+a) .

@ Examples: Sturzu qp/02 [ψ(s) = ∑k=–∞ exp(–k2/s2)]; Cvijović PRS(07) [polylogarithm]; Tsionskiy & Tsionskiy a1207-wd [comments on infinitely differentiable function of bounded support]; > s.a. Gaussians; Sigmoid.

Expansions and Operations on Functions > s.a. fourier analysis; Integral Transforms.
* Convolution: For two functions f and g on $$\mathbb R$$, the convolution is defined by

(f *g)(x):= $$\,_{\mathbb R}^~$$ dy f(xy) g(y) ;

The Fourier transform of the product of two functions is the convolution of the two Fourier transforms.

Generalizations > see distributions; Extrafunctions.

Online resources > see EquPlus [science and math equations in TeX, MathML, png-image and MathType format].