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 JCAP(16)-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 5 apr 2018