The horizontal line y = c is a horizontal asymptote of the function y = ƒ ( x ) if

In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function ƒ ( n ) defined on the set of natural numbers ( i. e. positive integers ) that " expresses some arithmetical property of n ."

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

Condition numbers can be defined for any function ƒ mapping its data from some domain ( e. g. an m-tuple of real numbers x ) into some codomain an n-tuple of real numbers ƒ ( x ), where both the domain and codomain are Banach spaces.

Conversely, if ƒ: C → C is a function which is differentiable when regarded as a function on R < sup > 2 </ sup >, then ƒ is complex differentiable if and only if the Cauchy – Riemann equations hold.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

So assume ƒ is differentiable at 0, as a function of two real variables from Ω to C. This is equivalent to the existence of two complex numbers and ( which are the partial derivatives of ƒ ) such that we have the linear approximation

It is possible to construct a continuous function satisfying the Cauchy – Riemann equations at a point, but which is not analytic at the point ( e. g., ƒ ( z ) =.

* If ƒ ( z ) is locally integrable in an open domain Ω ⊂ C, and satisfies the Cauchy – Riemann equations weakly, then ƒ agrees almost everywhere with an analytic function in Ω.

for n > 0 and some arbitrary but differentiable function ƒ ( x ) on the unit interval.

Then, using the periodic Bernoulli function P < sub > n </ sub > defined above and repeating the argument on the interval, one can obtain an expression of ƒ ( 1 ).

For example, the parent function y = 1 / x has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant.

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.

The function F ( x )

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

* Ai ( x ), the Airy function, a special function in mathematics

For curves given by the graph of a function, horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to Vertical asymptotes are vertical lines near which the function grows without bound.

Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞.

The line x = a is a vertical asymptote of the graph of the function

A common example of a vertical asymptote is the case of a rational function at a point x such that the denominator is zero and the numerator is non-zero.

For example, the function has a horizontal asymptote at y = 0 when x tends both to −∞ and +∞ because, respectively,

The relation between osmotic pressure and the Gibbs function may also be developed in an analogous way.

Thus, if public pressure sets the effective limit to the price that the industry may charge, this pressure is itself a function of the wage rate.

Extreme caution should be used, however, to avoid the conflicting usage of an index word or electronic switch which may result from the assignment of more than one name or function to the same address.

The policy may not be unique but the maximum value of P certainly is, and once the policy is specified this maximum can be calculated by ( 2 ) and ( 3 ) as a function of the feed state Af.

It may be that the objective function depends not only on Af but also on Af, as when the cost of the operating policy is considered.

`` Actually, the abuse of the process may have constituted a contempt of the Criminal court of Cook county, altho vindication of the authority of that court is not the function of this court '', said Karns, who is a City judge in East St. Louis sitting in Cook County court.

Today, in the era of the integrated musical when an individual song must contribute to the over-all development of the show, it is understandable that a song, no matter how excellent it may be on its own terms, is cut out because it does not perform the function required of it.

In practice, a bidirectional reflectance distribution function ( BRDF ) may be required to characterize the scattering properties of a surface accurately, although the albedo is a very useful first approximation.

Its size is determined by its function as a glycogen and fat storage unit, and may change with the seasons as these reserves are built or used up.

Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist.

The same function may have several different algorithms ".

Decomposition, simplification, or slowing of multijoint movement may also be an effective strategy that therapists may use to improve function in patients with ataxia.

In Greek practice the title or function of Abbot corresponds to a person who actually serves as the head of a monastery, although the title of the Archimandrite may be given to any celibate priest who could serve as the head of a monastery.

They may not administer Penance ( Reconciliation ), Anointing of the Sick ( Extreme Unction ), or function as an ordained celebrant or concelebrant of the Mass ( by virtue of their office and their training and institution, they may act, if the need arises, as altar servers, lectors, ushers, porters, or Eucharistic ministers of the Cup, and if need be, the Host ).

It is has been argued that Luke may be writing: a letter of apology ( traditionally a defense for one ’ s beliefs ), a letter of legitimation for Christian beliefs, a letter to equip the church to function amidst the Roman Empire, or a letter that is apolitical.

Some scholars believe that the apologetic view of Luke ’ s work is overemphasized and that it should not be regarded as a “ major aim of the Lucan writings .” While Munck believes that purpose of Luke ’ s work is not that clear-cut and sympathizes with other claims, he believes that Luke ’ s work can function as an apology only in the sense that it “ presents a defense of Christianity and Paul ” and may serve to “ clarify the position of Christianity within Jewry and within the Roman Empire .” Pervo disagrees that Luke ’ s work is an apology and even that it could possibly be addressed to Rome because he believes that “ Luke and Acts speak to insiders, believers in Jesus .” Freedman believes that Luke is writing an apology but that his goal is “ not to defend the Christian movement as such but to defend God ’ s ways in history .”

Functionalists like Monroe Beardsley argue that whether or not a piece counts as art depends on what function it plays in a particular context ; the same Greek vase may play a non-artistic function in one context ( carrying wine ), and an artistic function in another context ( helping us to appreciate the beauty of the human figure ).

These antidepressants typically have fewer adverse effects than the tricyclics or the MAOIs, although such effects as drowsiness, dry mouth, nervousness, anxiety, insomnia, decreased appetite, long-term weight gain and decreased ability to function sexually may occur.

Its function is to control the expulsion of feces, unwanted semi-solid matter produced during digestion, which, depending on the type of animal, may be one or more of: matter which the animal cannot digest, such as bones ; food material after all the nutrients have been extracted, for example cellulose or lignin ; ingested matter which would be toxic if it remained in the digestive tract ; and dead or excess gut bacteria and other endosymbionts.

In a way the rules of evidence can function to give a judge limited inquisitorial powers as the judge may exclude evidence he / she believes is not trustworthy or irrelevant to the legal issue at hand.