Several elementary functions which are defined via power series may be defined in any unital Banach algebra ; examples include the exponential function and the trigonometric functions, and more generally any entire function.

When α is an integer, the Bessel functions J are entire functions of x.

If x is held fixed, then the Bessel functions are entire functions of α.

Typical examples of entire functions are the polynomials and the exponential function, and any sums, products and compositions of these, including the error function and the trigonometric functions sine and cosine and their hyperbolic counterparts the hyperbolic sine and hyperbolic cosine functions.

Neither the natural logarithm nor the square root functions can be continued analytically to an entire function.

In a data stream, both text and control ( or formatting functions ) are interspersed allowing an entire screen to be " painted " as a single output operation.

On a national level, this is controversial in many countries due to technical mistrust of the security of the escrow arrangement ( due to a long history of less than adequate protection of others ' information by assorted organizations, public and private, even when the information is held only under an affirmative legal obligation to protect it from unauthorized access ), and to a mistrust of the entire system even if it functions as designed.

* In centralized peer-to-peer systems, a central server is used for indexing functions and to bootstrap the entire system.

An alternative approach was developed in 1929 by Friedrich Hund and Robert S. Mulliken, in which electrons are described by mathematical functions delocalized over an entire molecule.

This was the first attempt to describe many-electron systems on the basis of electronic density instead of wave functions, although it was not very successful in the treatment of entire molecules.

Host switches also perform OAM & P ( Operation, Administration, Maintenance, and Provisioning ) functions, including billing, for the entire cluster of the host and its remote switches.

Thus the set of functions, where g is any one-argument function, represents the entire set of functions in variables x, y that could have produced the x-partial derivative 2x + y.

These equations are in fact even valid for complex values of x, because both sides are entire ( that is, holomorphic on the whole complex plane ) functions of x, and two such functions that coincide on the real axis necessarily coincide everywhere.

The Paley – Wiener theorem relates growth properties of entire functions on C < sup > n </ sup > and Fourier transformation of Schwartz distributions of compact support.

In Middlesex County ( as in the entire Commonwealth of Massachusetts ) governmental functions such as property tax assessment and collection, public education, road repair and maintenance and elections are all conducted at the city and town level and not by county government.

The original proof uses hypergeometric functions and innovative tools from the theory of Hilbert spaces of entire functions, largely developed by de Branges.

He began the dialogue by having his wife announce that one does not invade people's homes without warning them that one is coming, and went on from that with the entire catalogue of his social gaucheries.

Christ's College was well represented that year in the ordo, and the name highest on the list from that college was Milton's, fourth in the entire university.

The entire middle section of The Walnut Trees is taken up with the life of Vincent Berger himself, whose fragmentary notes on his `` encounters with mankind '' are now conveyed by his son.

We congratulate the entire membership on its record of good legislation.

Moreover, the entire state Democratic hierarchy, from Gov. Brown on down to the county chairmen, also participates in this huge operation.

Most women, in this age of freezers, shop for the entire week on week-ends, when prices are lower.

He seems to have at least a few 30- and 50-megaton bombs on hand, since we cannot assume that he has exploded his entire stock.

Beginning in 1960, the matching requirements for the base allotment are being adjusted ( upward or downward, as required ) 25% a year, so that by 1963 the entire support allotment will be matched on the basis of a 40% pivot State share, with maximum and minimum State shares of 50% and 30%, respectively.

and that all this would have a serious adverse effect on the entire stock market and on general business activity.

Used in several sets of high reps once or twice each week it will not be long before your entire upper leg takes on a razor-sharp definition in which the muscles look like wire cables writhing and twisting under the skin!!

Each generator, **yl, of Af is also exceptional, for each is transformed into the entire congruence of secants of the curve into which that generator is transformed by the point involution on Q.

There was no extra horse so it was left to his comrades who, though numbering in the fifties, had stood around on the hillside nearby without firing a shot during the entire attack.

The appeal of the suburb is particularly strong for heavy industry, which must move bulky objects along a lengthy assembly line and wants enough land area to do the entire job on one floor.

An extended cold spell caused ice to build up on the aerator which was mounted on a floating platform and caused the entire platform to sink lower in the water.

Its entire complement of non-commissioned officers on the platoon level had departed as cadre for another unit, and its vehicles were still those used in the drive across Luzon in World War 2.

In any case, anyone who fails to make significant distinction between primary and secondary applications of economic pressure would in principle already have justified that use of economic boycott as a means which broke out a few years ago or was skillfully organized by White Citizens' Councils in the entire state of Mississippi against every local Philco dealer in that state, in protest against a Philco-sponsored program over a national TV network on which was presented a drama showing, it seemed, a `` high yellow gal '' smooching with a white man.

He sucked in his breath and kept quiet while Killpath laid down the sheet again, wound the gold-wire stems of his glasses around his ears and then, eying the report as it lay before him on the desk, intoned, `` Acting Lieutenant Gunnar Matson one failed to see that the station keeper was properly relieved two absented himself throughout the entire watch without checking on the station's activities or the whereabouts of his section sergeants three permitted members of the Homicide Detail of the Inspector's Bureau to arrogate for their own convenience a patrolman who was thereby prevented from carrying on his proper assignment four failed to notify the station commander Acting Captain O. T. Killpath of a homicide occurring in the district five frequented extralegal establishments known as after-hours spots for purposes of an unofficial and purportedly social nature and six '' -- he leaned back and peeled off his glasses `` -- failed to co-operate with the Acting Captain by returning promptly when so ordered.

The entire headquarters collection is available to the patrons of all members on interlibrary loans.

So we are the more prepared to give Parker the credit for having taken the right side in an unnecessary controversy, to salute his courage, and to pass on, happily forgetting both him and the entire episode.

After implementing the Emancipation Proclamation, which did not apply to every state, Lincoln increased pressure on Congress to outlaw slavery throughout the entire nation with a constitutional amendment.

While 2 November remained the liturgical celebration, in time the entire month of November became associated in the Western Catholic tradition with prayer for the departed ; lists of names of those to be remembered being placed in the proximity of the altar on which the sacrifice of the mass is offered.

The most obvious of these is the quadratic complex of tangents to Q, each line of which is transformed into the entire pencil of lines tangent to Q at the image of the point of tangency of the given line.

Computational fluid dynamics was started as an effort to solve for flow properties around complex objects and has rapidly grown to the point where entire aircraft can be designed using a computer, with wind-tunnel tests followed by flight tests to confirm the computer predictions.

* more recently, compare entire genomes, which permits the study of more complex evolutionary events, such as gene duplication, horizontal gene transfer, and the prediction of factors important in bacterial speciation,

Higher clock rates in increasingly complex CPUs make it more difficult to keep the clock signal in phase ( synchronized ) throughout the entire unit.

Almost the entire Casablanca waterfront is under development, mainly the construction of huge entertainment centres between the port and Hassan II Mosque, the Anfa Resort project near the business, entertainment and living centre of Megarama, the shopping and entertainment complex of Morocco Mall, as well as a complete renovation of the coastal walkway.

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane.

As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere ( complex plane and the point at infinity ) is constant.

Thus any non-constant entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire function.

Specifically, by the Casorati – Weierstrass theorem, for any transcendental entire function f and any complex w there is a sequence with, is necessarily a polynomial, of degree at least n.

This last function is equal to the inverse cosine trigonometric function in the entire complex domain.

The Anstoss thus provides the essential impetus that first sets in motion the entire complex train of activities that finally result in our conscious experience both of ourselves and others as empirical individuals and of the world around us.

There is in fact no complex number z for which, and hence the reciprocal gamma function is an entire function, with zeros at z = 0, − 1, − 2 ,.... We see that the gamma function has a local minimum at where it attains the value.

The company will lease most of the antennas for at least three years with the option to buy the entire complex in the future.

A holomorphic function whose domain is the whole complex plane is called an entire function.

Procedures and training are significantly more complex as a pilot must demonstrate competency in conducting an entire cross-country flight in IMC conditions, while controlling the aircraft solely by reference to instruments.

After the last retreat of the ice sea level changes and the isostatic rise of land makes charting post glacial coastlines a complex task but the resultant clifflines behind raised beaches are a prominent feature of the entire coastline.

The entire image for the frame must be output when the visual difference between the two frames is so great that representing the new image incrementally from the previous frame would be more complex and would require even more bits than reproducing the whole image.

# The entire complex ( including the bound Nut site on the mRNA ) continues transcription, and can skip through termination sequences.

The entire nuclear pore complex ( NPC ) has a diameter of about 120 nanometers, the diameter of the opening ( functional diameter ) is about 9 nanometers wide and its " depth " is about 200 nanometers.

Critics see it as an upgrade to the entire nuclear weapons complex to support the production and maintenance of the new generation of nuclear weapons.

On 17 December 2007, Daimler announced that they were selling their entire complex of 19 buildings at Potsdamer Platz to SEB Asset Management, a Frankfurt-based subsidiary of the Swedish banking group SEB.

The roots of the theory go back to the result of Émile Picard in 1879, showing that a non-constant complex-valued function which is analytic in the entire complex plane assumes all complex values save at most one.