Bessel function (12669 bytes)
1: ...Bessel]], are canonical solutions ''y''(''x'') of Bessel's [[differential equation]]:
7: ...o that the Bessel functions are mostly [[analytic function]]s of α).
11: ...n. (For cylindrical problems, one obtains Bessel functions of integer order α = ''n''; for spherical ...
17: Bessel functions also have useful properties for other problems, ...
23: === Bessel functions of the first and second kind ===

Friedrich Bessel (4195 bytes)
1: ... in Königsberg (now [[Kaliningrad]], [[Russia]]). Bessel was a contemporary of [[Carl Friedrich Gauss|Carl...
3: Bessel was the son of a civil servant, and at the age of...
9: ...ccurately measure a stellar parallax. In [[1838]] Bessel won the "race", announcing that [[61 Cygni]] had ...
11: ...ll as helping determine the parallax of 61 Cygni, Bessel's precise measurements allowed him to notice devi...
13: Despite lacking a university education, Bessel was a major figure in astronomy during his lifeti...

Bessel (crater) (1826 bytes)
7: eponym=[[Friedrich Wilhelm Bessel|Friedrich W. Bessel]]}}
8: ... crosses the mare from north to south, passing by Bessel crater along the western side.
12: ...e side of the crater mid-point that is closest to Bessel
16: !width="25%" style="background:#eeeeee;" |Bessel
43: * Bessel A — ''See'' [[Sarabhai crater]].

Bessel's inequality (1138 bytes)
1: In [[mathematics]], '''Bessel's inequality''' is a statement about the coeffici...
8: Bessel's [[inequality]] tells us that this [[series (mat...
12: {{planetmath|title=Bessel inequality|id=3089}}
13: [[Category:Functional analysis]]

Z function (5258 bytes)
1: ...he '''Z function''' is a [[function (mathematics)|function]] used for studying the
2: [[Riemann zeta function]] along the critical line where the real part of ...
3: ... one-half. It is also called the Riemann-Siegel Z function, the
4: ... the zeta function and the [[Riemann-Siegel theta function]] by
7: ...o zeros off the critical line of the Riemann zeta function in its critical strip.

One way function (2865 bytes)
1: ...[cryptography]] rests on the existence of one way functions.
3: Formally, a one-way function is a [[computable function]] <math>f</math> with the following properties:
8: ...f NP) implies the existence of one-to-one one-way functions; see [[UP (complexity)|UP]] for more.
10: It is known that the existence of one-way functions implies the existence of many other useful crypt...
12: *Pseudorandom function families;

How to specify a function (3172 bytes)
1: In [[mathematics]], '''[[function (mathematics)|function]]s''' can be specified in several ways.
2: In practice, most functions that relate (combinations of) numbers with numbe...
3: For example, the ''dist'' function can be specified with
9: ... is called an '''explicit function''', as it is a function being stated explicitly.
11: ...ifying the set of [[ordered pair]]s that form the function's graph by either enumerating it or specifying it...

Function (1448 bytes)
1: ... or evolutionary success. The common feature of a function is that it produces some kind of result.
3: '''Function''' may refer to:
6: ...ept of a '''[[function (mathematics)|mathematical function]]'''
7: ** a '''[[functional (mathematics)|functional]]'''
9: ...g language]], often analogous with a mathematical function;

L-function (3495 bytes)
3: ...n it, broad generalisations of the [[Riemann Zeta function]] and the L-series for a [[Dirichlet character]] ...
5: === L-functions ===
7: ...te series for the Riemann zeta-function), and the function in the complex plane that is its analytic continu...
9: ...L-function here includes many known types of zeta-function.
13: ...e can list characteristics of known examples of L-functions that one would wish to see generalised:

Eta function (160 bytes)
1: There are two distinct functions called '''eta function''' in [[number theory]]:
3: * The [[Dirichlet eta function]]
4: * The [[Dedekind eta function]]

Loss function (2754 bytes)
1: ...y]] and [[economics]], a '''loss function''' is a function that maps an [[event (probability theory)|event]]...
3: Loss functions are typically expressed in monetary terms though...
5: ... functions are complementary to [[utility|utility functions]] which represent benefit and satisfaction. Typi...
9: ...h [[probability density function]] ''f'' and loss function ''&lambda;'', the ''expected loss'' (sometimes kn...
15: ==Loss functions in Bayesian statistics==

Synchrotron function (359 bytes)
1: In [[mathematics]] the '''synchrotron functions''' are defined as follows (for ''x'' &ge; 0):
3: * First synchrotron function
7: * Second synchrotron function
11: ...re ''K''_''j'' is the modified [[Bessel function]] of the second kind.

Transport function (165 bytes)
1: In [[mathematics]], the '''transport functions''' ''J''(''n'',''x'') are defined by

Divisor function (1653 bytes)
1: In [[mathematics]] the '''divisor function''' &sigma;_''a''(''n'') is defined as t...
5: ...or the number of divisors of ''n''. The '''sigma function''' &sigma;(''n'') is
15: ...nerally, the divisor function is [[multiplicative function|multiplicative]], but not [[completely multiplica...
23: This function is the one used to recognize [[perfect number]]s ...
33: Two [[Dirichlet series]] involving the divisor function are:

Fitness function (991 bytes)
1: ...s function''' is a particular type of [[objective function]] that quantifies the optimality of a solution (t...
3: Another way of looking at fitness functions is in terms of a [[fitness landscape]], which sh...
5: An ideal fitness function correlates closely with the algorithm's goal, and...

Production function (11657 bytes)
2: ... state of available technology. Unique production functions can be constructed for every production technolo...
4: Alternatively, a production function can be defined as the specification of the minimu...
6: ...lationship is non-monetary, that is, a production function relates physical inputs to physical outputs. Pric...
8: ==The production function as an equation==
9: ... its most general mathematical form, a production function is expressed as:

Inline function (6182 bytes)
1: ... body of the function in every context where that function is used.
5: ... case. It also has a space benefit for very small functions, and is an enabling transformation for other [[o...
7: ...uted functions, to be exploited in choosing which functions to inline.
9: ...on in every module that uses it, whereas ordinary functions must be defined in only a single module. This fa...
14: ...ven check that arguments are well-formed, whereas function calls usually do.

Mertens function (1255 bytes)
1: In [[number theory]], the '''Mertens function''' is
5: where &mu;(k) is the [[M&ouml;bius function]].
7: ... no ''x'' where the absolute value of the Mertens function exceeds the square root of ''x''. The Mertens co...
10: *Values of the Mertens function for the first 50 ''n'' are given by [http://www.r...
11: *Values of the Mertens function for the first 2500 ''n'' are given by [http://www...

Survivor function (1319 bytes)
1: ...mmon in [[engineering]] while the term ''survivor function'' is used in a broader range of applications, inc...
5: ...n;)</nowiki>. Its ''survivor-'', or ''reliability-function'' is:
11: Every survivor function ''R''(''t'') is [[monotone decreasing]].
22: *[[Cummulative hazard function]]

Weierstrass function (704 bytes)
1: ... property is both stable and generic. Weierstrass functions are defined by
9: The following graphs display the function <math>f(x)=\sum_{n=0}^\infty (1/2)^n\cos(20^n\pi ...

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