If a complex function = is holomorphic, then u and v have first partial derivatives with respect to x and y, and satisfy the Cauchy – Riemann equations:

A simple converse is that if u and v have continuous first partial derivatives and satisfy the Cauchy – Riemann equations, then ƒ is holomorphic.

A more satisfying converse, which is much harder to prove, is the Looman – Menchoff theorem: if ƒ is continuous, u and v have first partial derivatives ( but not necessarily continuous ), and they satisfy the Cauchy – Riemann equations, then ƒ is holomorphic.

One way of proving that a given disease is " infectious ", is to satisfy Koch's postulates ( first proposed by Robert Koch ), which demands that the infectious agent be identified only in patients and not in healthy controls, and that patients who contract the agent also develop the disease.

Having found for the plaintiff for a violation of the first law, the court then has the discretion to exercise judicial economy and refuse to make a decision on the remaining two claims, on the grounds that the finding of one violation should be sufficient to satisfy the plaintiff.

The first four functions help the child to satisfy physical, emotional and social needs.

Towards the latter part of the Middle Ages in Europe, both the State-the State would use the instrument of confiscation for the first time to satisfy a debt-and the Church-the Church succeeded in acquiring immense quantities of land-were allied against the village community to displace the small landlord and they were successful to the extent that today, the village has become the ideal of the individualist, a place in which every man " does what he wills with his own.

Ultimately, the FIFA stepped in to satisfy the players ' requirements and the first boycott of a FIFA World Cup game never happened.

Sets of these cardinalities satisfy the first three characterizations above but not the fourth characterization.

The first large project Wren designed, the Chelsea Hospital ( 1682 – 92 ), does not entirely satisfy the eye in this respect, but met its brief with distinction and such success that even in the 21st century it fulfills its original function.

He is first usually required to check, and certify, the regularity of the relationship of the Lodge-and be able to satisfy that Lodge of his regularity of membership.

The first bank was opened in 1868 by W. M. Johnson, who was forced two years later to make an assignment of all that he had to satisfy his creditors.

To satisfy Southern opposition to the general principle of Federally supported internal improvements, the land grants would first be transferred to the appropriate state or territorial government, which would oversee the final transfer to private developers.

The company was unable to fully satisfy either the casual or serious model building market, and was one of the first companies to abandon model kits when the hobby started its decline, selling AMT to the Ertl Company and shutting down its own kit division less than twenty years after starting it.

If these requirements are not met, it's not possible to interpret the wavefunction as a probability amplitude ; the values of the wavefunction and its first order derivatives may not be finite and definite ( with exactly one value ), i. e. probabilities can be infinite and multiple-valued at any one position and time-which is nonsense, as it does not satisfy the probability axioms.

Otto had already submitted a written statement, on 21 February 1958, that he and his family would renounce all privileges formerly entitled a member of the House of Habsburg, but this first declaration did not satisfy the requirements of the Habsburg Law, which stated that Otto and other descendants of Charles could only return to Austria if they renounced all royal claims and accepted the status of private citizens.

As money comes in from the debtor to the creditor, it is first applied to satisfy any additional costs and interest, and only then, the principal balance of the judgment.

* New releases or high-demand titles are not always immediately available through ILL because most libraries need to satisfy local demand first.

It can be shown that C must be zero and in order to satisfy the Jacobi identity, A, B and D must satisfy a non-linear first order differential equation.

It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926 while searching for solutions of the Schrödinger equation that satisfy the correspondence principle.

By 1940, he had changed the spelling of his name from " Willis " to " Wyllis " ( to satisfy " his wife's numerological inclinations ") and lived mainly in the New York City area where he worked on a number of radio programs, the most important of which was probably Edward M. Kirby's popular and acclaimed government propaganda series, The Army Hour, which Cooper wrote, produced and directed for its first year.

The Porsche 959 was a sports car manufactured by Porsche from 1986 to 1989, first as a Group B rally car and later as a legal production car designed to satisfy FIA homologation regulations requiring that a minimum number of 200 street legal units be built.

A second verse was added to Lothario's aria in the first act (" Fugitif et tremblant " in the French version ), and in the second act, a rondo-gavotte for Frédéric (" Me voici dans son boudoir ") was devised using the music of the entr ' acte preceding that act, to satisfy Mme Trebelli-Bettini, who was discomfited by having to take on a role originally written for buffo tenor.

Telling a person he is " under arrest " is sufficient to satisfy this requirement even though the person may not be otherwise physically restrained.

Students are given a variety of class choices to fulfill the seminar requirement, and to satisfy the first-year writing requirement.

For example, in Hanson v. Denckla, the Court proclaimed the " unilateral activity of those who claim some relationship with a nonresident cannot satisfy the requirement of contact with the forum State.

In most cases, transferred intent, which occurs when the defendant intends to injure an individual but actually ends up injuring another individual, will satisfy the intent requirement.

In order to satisfy this requirement, it turns out that in addition to the black hole interior region which particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region which allows us to extrapolate the trajectories of particles which an outside observer sees rising up away from the event horizon.

The probability measure function must satisfy a simple requirement: the probability of a union of two ( or countably many ) disjoint events must be equal to the sum of probabilities of each of these events.

Robison, Schmid, and Siles ( 2002 ) reviewed various definitions of social capital and concluded that many did not satisfy the formal requirement of a definition.

Peaches grow very well in a fairly limited range, since they have a chilling requirement that low altitude tropical areas cannot satisfy.

In tropical and equatorial latitudes, such as Ecuador, Colombia, Ethiopia, India and Nepal, they grow at higher altitudes that can satisfy the chilling requirement.

Upon graduation, students must demonstrate a proficiency in a second language at an intermediate level, satisfy a quantitative reasoning requirement, and complete a senior individualized project which may take the form of a thesis, an artistic performance, or any other work-intensive project of a student's choosing.

The integrating provinces of Castile – La Mancha opposed such a special status, and after considering other options — like its inclusion to the community of Castile and León or its constitution as an entity similar to a federal district — it was decided that the province of Madrid would become a single-province autonomous community by appealing to the 144th article of the constitution, whereby the Parliament can authorize the creation of an autonomous community, even if it did not satisfy the requirement of having a distinct historical identity, if it was the " nation's interest ".

Ali, the bumiputra, is included solely to satisfy this requirement, and Baba ( the non-bumiputra ) pays Ali a certain sum in exchange.

To qualify as a magma, the set and operation must satisfy the following requirement ( known as the magma axiom ):

Professional degrees such as the MArch ( Master of Architecture ) can last to three and a half years to satisfy professional requirement to be an architect.

After scriptwriting on two productions, Powell entered into a partnership with American producer Jerry Jackson in 1931 to make " quota quickies ", Powell began to direct hour-long films needed to satisfy a legal requirement that British cinemas screen a certain quota of British films.

In recognizing Mutai's mark as the " fastest Marathon ever run ", the International Association of Athletics Federations noted that the performance was not eligible for world record status given that the course does not satisfy rules regarding elevation drop and start / finish separation ( the latter requirement being intended to prevent advantages gained from a strong tailwind, as was the case in 2011 ).

Essentially, Calabi – Yau manifolds are shapes that satisfy the requirement of space for the six " unseen " spatial dimensions of string theory, which may be smaller than our currently observable lengths as they have not yet been detected.

The awkwardness of the of Anaheim suffix, appended to satisfy a contractual requirement for Anaheim to be included in the team's name, also caused the new name to become the subject of ridicule nationally.

In order to satisfy this requirement, it turns out that in addition to the black hole interior region which particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region which allows us to extrapolate the trajectories of particles which an outside observer sees rising up away from the event horizon.

Mikhail Gromov, however, made the important observation that symplectic manifolds do admit an abundance of compatible almost complex structures, so that they satisfy all the axioms for a Kähler manifold except the requirement that the transition functions be holomorphic.

Although the vast majority of law graduates fulfill the articled clerkship requirement by articling ( i. e. working and learning ) in a law firm, a government's legal department, a corporation's ( in house ) legal department, a community legal clinic or some other type of non-profit organization involved in legal work, a small minority of law graduates ( with exceptional academic records ) satisfy the articled clerkship requirement by undergoing what is commonly called clerkship with a specific courthouse and under the supervision of a judge instead of working in a more " lawyer-type environment " under the supervision of a lawyer called a " principal ".

However, aside from a common lower division religious studies course ( introduction to biblical literature ), students can take a variety of different courses to fulfill these requirements ( for example, a course on U. S. History and a course on Psychology would satisfy the social science requirement ).