The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler – Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

His work is notable for the use of the zeta function ζ ( s ) ( for real values of the argument " s ", as are works of Leonhard Euler, as early as 1737 ) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π ( x )/( x / ln ( x )) as x goes to infinity exists at all, then it is necessarily equal to one.

Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ ( s ) is non-zero for all complex values of the variable s that have the form s = 1 + it with t > 0.

The values of the Riemann zeta function at even positive integers were computed by Euler.

The following are the most commonly used values of the Riemann zeta function.

This equation relates values of the Riemann zeta function at the points s and.

For s an even positive integer, the product sin ( πs / 2 ) Γ ( 1 − s ) is regular and the functional equation relates the values of the Riemann zeta function at odd negative integers and even positive integers.

This image shows a plot of the Riemann zeta function along the critical line for real values of t running from 0 to 34.

The series on the right is just a series representation of the Riemann zeta function, but it only converges for values of s-n that are greater than unity.

where ζ ( s, q ) is the Hurwitz zeta ; thus, in a certain sense, the Hurwitz zeta generalizes the Bernoulli polynomials to non-integer values of n.

* Grothendieck expressed the zeta function in terms of the trace of Frobenius on l-adic cohomology groups, so the Weil conjectures for a d-dimensional variety V over a finite field with q elements depend on showing that the eigenvalues α of Frobenius acting on the ith l-adic cohomology group H < sup > i </ sup >( V ) of V have absolute values | α |= q < sup > i / 2 </ sup > ( for an embedding of the algebraic elements of Q < sub > l </ sub > into the complex numbers ).

pointed out that a generalization of Rankin's result for higher even values of k would imply the Ramanujan conjecture, and Deligne realized that in the case of zeta functions of varieties, Grothendieck's theory of zeta functions of sheaves provided an analogue of this generalization.

For rational values of ( that is, for for some integers p and q ), the function can be understood to represent a periodic orbit of an element in the cyclic group, and thus can be expressed as a simple sum involving the Hurwitz zeta function.

The functional equation relates values of the zeta on the left-and right-hand sides of the complex plane.

The Hurwitz zeta function occurs in a number of striking identities at rational values.

Kazuya Kato refers to the elements in an Euler system as " arithmetic incarnations of zeta " and describes the property of being an Euler system as " an arithmetic reflection of the fact that these incarnations are related to special values of Euler products ".

The Dedekind zeta function satisfies a functional equation relating its values at s and 1 − s. Specifically, let Δ < sub > K </ sub > denote discriminant of K, let r < sub > 1 </ sub > ( resp.

Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode ( at least conjecturally ) important arithmetic data of the field K. For example, the analytic class number formula relates the residue at s = 1 to the class number h ( K ) of K, the regulator R ( K ) of K, the number w ( K ) of roots of unity in K, the absolute discriminant of K, and the number of real and complex places of K. Another example is at s = 0 where it has a zero whose order r is equal to the rank of the unit group of O < sub > K </ sub > and the leading term is given by

Examples of Lambert series in this form, with, occur in expressions for the Riemann zeta function for odd integer values ; see Zeta constants for details.

No such representation in terms of π is known for the so-called odd zeta constants, the values ζ ( 2n + 1 ) for positive integers n. It has been conjectured that the ratios of these quantities

In some neighborhood of an isolated tangent point in the f-plane, say Af, the function Af is either double-valued or has no values defined, except at the tangent point itself, where it is single-valued.

In the second category the function Af has no values defined in a neighborhood Af.

The number of ordinary values of the function f{t} at T will be called its multiplicity at T.

The values Af are the ordinary values at Af of a multi-valued function g{t} which has components corresponding to those of f{t}.

It differs from a digital signal, in which a continuous quantity is represented by a discrete function which can only take on one of a finite number of values.

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.

Positive h and k values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.

In statistics, the autocorrelation of a random process describes the correlation between values of the process at different points in time, as a function of the two times or of the time difference.

Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

For small values of m like 1, 2, or 3, the Ackermann function grows relatively slowly with respect to n ( at most exponentially ).

To emphasize that they are being thought of as functions rather than sequences, values of an arithmetic function are usually denoted by a ( n ) rather than a < sub > n </ sub >.

This would correspond to the mathematical notion of the support of a function ( i. e., the total " length " of values for which the function is nonzero ).

where Γ ( z ) is the gamma function, a generalization of the factorial function to non-integer values.

Another definition of the Bessel function, for integer values of, is possible using an integral representation:

* Procedures ( subroutines not returning values ) are a special case of function, with a dummy return type.

The true and final aim of thought coercion is to induce a change in the victim ’ s objective function itself, i. e. the basic set of values and rules by which the victim determines his or her own choice among the alternatives of any feasible set.

Currying is similar to the process of calculating a function of multiple variables for some given values on paper.

:" It was stated ... that ' a function is effectively calculable if its values can be found by some purely mechanical process.

In this way, numerical values for the relative efficacy of various detergent formulations can be obtained by measuring the reflectance ( whiteness ) of the cloth swatches before and after washing.

Even in the neighborhood of the Earth, where information has been obtained both directly and indirectly, the derived flux values vary by at least four orders of magnitude.

The increment and the fluxion are obtained for a particular time that determines the values of the quantities on the righthand sides of the above rules.

The values obtained are compared with those for TNT.

In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable.

By choosing different values for the parameter m, the following types of means are obtained:

The correction is named after Victor Goldschmidt who obtained the numerical values quoted above.

The terminology is also applied to indirect measurements — that is, values obtained by a computational procedure from observed data.

The CIPM have categorised it as a " non-SI unit " because units values in SI units must be obtained experimentally.

* In 2006, in the 8th edition of the formal definition of SI, the CIPM cataloged the dalton alongside the unified atomic mass unit as a " Non-SI units whose values in SI units must be obtained experimentally: Units accepted for use with the SI ".

Using a mathematical formulation, let ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub >) be the input sequence of n distinct values over the finite field F ; then the codebook C created from the tuplets of values obtained by evaluating every polynomial ( over F ) of degree less than k at each x < sub > i </ sub > is

The result values are said to be commensurate if they are obtained ( in distinct experimental trials ) according to the same reproducible experimental description and procedure.

* All the rules that apply are invoked, using the membership functions and truth values obtained from the inputs, to determine the result of the rule.

The numeric answer can be obtained by substituting the above values into this formula.

The equation obtained for is a cubic whose numerical solution yields the following values for the pH and the different species concentrations:

The standard enthalpy of combustion is the enthalpy change when one mole of a reactant completely burns in oxygen under standard thermodynamic conditions ( although experimental values are usually obtained under different conditions and subsequently adjusted ).

The first mention of equal temperament related to Twelfth root of two in the West appeared in Simon Stevin's unfinished manuscript Van de Spiegheling der singconst ( ca 1605 ) published posthumously three hundred years later in 1884 ; however, due to insufficient accuracy of his calculation, many of the numbers he obtained were off by one or two units from the correct values.

The fact that the values obtained both by electrical and purely mechanical means were in agreement to at least one order of magnitude was, to Joule, compelling evidence of the reality of the convertibility of work into heat.

By weighting and integrating individual returns ( such as through likelihood ratio based detectors ), detection can be optimised to place more emphasis on stronger returns obtained from certain monostatic or bistatic radar cross section values, or from favourable propagation paths, when making a decision as to whether a target is present.

This is done by defining a sequence of value functions V < sub > 1 </ sub >, V < sub > 2 </ sub >, ..., V < sub > n </ sub >, with an argument y representing the state of the system at times i from 1 to n. The definition of V < sub > n </ sub >( y ) is the value obtained in state y at the last time n. The values V < sub > i </ sub > at earlier times i = n − 1, n − 2, ..., 2, 1 can be found by working backwards, using a recursive relationship called the Bellman equation.

The city has not obtained its own zip code even though, based on the zip code designation, some sales taxes paid by Newcastle residents go to Renton instead of Newcastle as tax revenues ( among other perceived detrimental impacts related to mail service, insurance rates, property values, and credit ratings ).

Regardless of the type of tool used, the quantitative values obtained can be used in combination with anticipated reductions ( for the targeted code ) to estimate a relative or absolute overall saving.