Functions| Functions |
In General > s.a. analysis;
functional analysis.
* Idea: A map f :
M → K from a manifold M to
a field K (usually, K = R or
C, and M may be R or C too).
Polynomials > s.a. Chebyshev,
Hermite,
Jack, Laguerre, legendre
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.
* 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 C separating z1,
..., zm from w1,
..., wn.
@ General references: Landau NAMS(87) [factoring];
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-in.
@ Other types: Edwards BAMS(09) [solvable].
Other Types > s.a. analytic and harmonic
functions; L-functions; Meromorphic;
series; summations.
* Almost periodic: A function of the form
f(x) = 
j=1n cj exp{i kjx}
,
for some integer n and real numbers cj, kj; They form an algebra and
can
be endowed with the sup norm, and is naturally isomorphic to the algebra
of
continuous functions on the Bohr compactification of R.
* 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] → 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 
[0,1],
the values of f inside the interval satisfy f(x +
(1–) y) 
f(x)
+ (1–) f(y)
.
* Convex: A function f :
[a, b] → 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 [0,1],
the values of f inside the interval satisfy f(x +
(1–) y) 
f(x)
+ (1–) f(y)
.
* Positive pure frequency
function: One of the form F(x, t)
= f(x) exp{–i
t},
or satisfying 
t f = –i f,
for > 0.
* Positive frequency function: One that can be extended to an analytic
function in the lower complex t-plane.
> Special
functions:
see Airy; bessel; Elliptic; Gamma; Hypergeometric; Jost; Mathieu; Struve; Whittaker; Zeta
Function; spherical
harmonics.
Examples and Properties > s.a. Hyperbolic
Functions; trigonometry.
* Other examples:
Cinfty function
of compact support:
f(x):= exp{–(x–x0)2/[(x–x0)2–h2]}
, 
(x):=
exp{–1/x2(a–x)2}
for x (0, a),
and 0 otherwise;
Cinfty function vanishing
for x 0 and equal
to 1 for x a:
(x):= C–1 
–inftyx (x')
dx' , C:=
R (x)
dx ;
Cinfty function on Rn,
of compact support, equal to 1 in a square box
xi (
i, 
i):
g(x1, ..., xn):= 
i=1n (xi–i+a)
(i–xi+a)
.
@ Examples: Sturzu qp/02 [(s)
= k=–inftyinfty exp(–k2/s2)];
Cvijovic PRS(07) [polylogarithm]; > s.a. Gaussians.
Expansions and Operations on Functions > s.a. fourier
analysis; Integral
Transforms.
* Convolution: For two functions f and g on R, the convolution
is defined by
(f *g)(x):= R dy f(x–y)
g(y) ;
The Fourier transform of the product of two functions is the convolution of the two Fourier transforms.
Related Concepts > see distributions [generalized functions]; Germ of a Function; Stirling's Formula [asymptotic behavior].
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 14
nov 2008