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*(*z*_{1},
..., *z*_{m}, *w*_{1},...,
*w*_{n}), separately of
degree 1 in each of its *m* + *n *arguments, such that *P*(*z*_{1},
..., *w*_{n}) ≠ 0 whenever
there is a circle in \(\mathbb C\) separating *z*_{1},
..., *z*_{m} from *w*_{1},
..., *w*_{n}.

@ __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* < *x*_{1}
< *x*_{2} < ... < *x*_{n–1}
< *b* of the interval, the sum of the variations |*f*(*x*_{i})
– *f*(*x*_{i–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*(*λ**x* + (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*(*λ**x*
+ (1–*λ*)* y*) ≥ *λ* *f*(*x*) +
(1–*λ*) *f*(*y*) .

* __Positive pure
frequency function__: One of the form *F*(*x*, *t*)
= *f*(*x*) exp{–i*ωt*}, or satisfying \(\cal L\)_{t
}*f* = –i*ω* *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{–(*x*–*x*_{0})^{2}/[(*x*–*x*_{0})^{2}–*h*^{2}]}
, *ψ*(*x*):= exp{–1/*x*^{2}(*a*–*x*)^{2}} for
*x* ∈ (0,* a*), 0 otherwise;

C^{∞} function vanishing for *x* ≤
0 and equal to 1 for *x* ≥ *a*,

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

C^{∞} function on \(\mathbb R\)^{n},
of compact support, equal to 1 in a square box *x*_{i}
∈ (*α*_{i}, *β*_{i}),

*g*(*x*_{1},
..., *x*_{n}):= ∏_{i=1}^{n}
*χ*(*x*_{i}–*α*_{i}+*a*)
*χ*(*β*_{i}–*x*_{i}+*a*)
.

@ __Examples__: Sturzu qp/02
[*ψ*(*s*) = ∑_{k=–∞}^{∞}
exp(–*k*^{2}/*s*^{2})];
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*):=
∫\(\,_{\mathbb R}^~\) d*y* *f*(*x*–*y*)
*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].

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send feedback and suggestions to bombelli at olemiss.edu – modified 11 aug
2016