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That is, a holomorphic function f has derivatives of every order at each point a in its domain, and it coincides with its own Taylor series at a in a neighborhood of a.
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is and holomorphic
Consequently, we can assert that a complex function f, whose real and imaginary parts u and v are real-differentiable functions, is holomorphic if and only if, equations ( 1a ) and ( 1b ) are satisfied throughout the domain we are dealing with.
The reason why Euler and some other authors relate the Cauchy – Riemann equations with analyticity is that a major theorem in complex analysis says that holomorphic functions are analytic and viceversa.
This means that, in complex analysis, a function that is complex-differentiable in a whole domain ( holomorphic ) is the same as an analytic function.
If is continuous in an open set Ω and the partial derivatives of ƒ with respect to x and y exist in Ω, and satisfies the Cauchy – Riemann equations throughout Ω, then ƒ is holomorphic ( and thus analytic ).
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane.
A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.
The existence of a complex derivative is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series.
The term analytic function is often used interchangeably with “ holomorphic function ”, although the word “ analytic ” is also used in a broader sense to describe any function ( real, complex, or of more general type ) that is equal to its Taylor series in a neighborhood of each point in its domain.
The fact that the class of complex analytic functions coincides with the class of holomorphic functions is a major theorem in complex analysis.
A holomorphic function whose domain is the whole complex plane is called an entire function.
If ƒ is complex differentiable at every point z < sub > 0 </ sub > in an open set U, we say that ƒ is holomorphic on U. We say that ƒ is holomorphic at the point z < sub > 0 </ sub > if it is holomorphic on some neighborhood of z < sub > 0 </ sub >.
is and function
Unconcerned with the practical function of his actions, the dancer is engrossed exclusively in their `` motional content ''.
This is the primary function of the imagination operating in the absence of the original experiential stimulus by which the images were first appropriated.
But because it is the function of the mind to turn the one into the other by means of the capacities with which words endow it, we do not unwisely examine the type of distinction, in the sphere of politics, on which decisions hang.
`` Mr. Gross, your report says that ' our function is investigative and advisory and does not in any way derogate from or prejudice Mr. Bang-Jensen's rights as a staff member.
It is, however, a disarming disguise, or perhaps a shield, for not only has Mercer proved himself to be one of the few great lyricists over the years, but also one who can function remarkably under pressure.
Men seem almost universally to want a sense of function, that is, a feeling that their existence makes a difference to someone, living or unborn, close and immediate or generalized.
And there is one other point in the Poetics that invites moral evaluation: Aristotle's notion that the distinctive function of tragedy is to purge one's emotions by arousing pity and fear.
It is possible that international organization will ultimately supplant the multi-state system, but its proper function for the immediate future is to reform and supplement that system in order to render pluralism more compatible with an interdependent world.
A primary function is the operation of a Government Bid Center, which receives bids daily from the Federal Government's principal purchasing agencies.
The education function of the Institute is carried on by the staff in the departments of pathology and its consultants.
The responsibility for taking the initiative in generating ideas is that of every officer in the Department who has a policy function, regardless of rank.
This function is staffed by engineers chosen for their technical competence and who have the title, member of the technical staff.
If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.
This is interesting for it combines both the thermodynamic concept of a minimum Gibbs function for equilibrium and minimum mechanical potential energy for equilibrium.
The concept of the strain energy as a Gibbs function difference Af and exerting a force normal to the shearing face is compatible with the information obtained from optical birefringence studies of fluids undergoing shear.
A proton magnetic resonance study of polycrystalline Af as a function of magnetic field and temperature is presented.
Within certain wide limits anatomy dictates function and, if one is permitted to speculate, potential pathology should be included in this statement as well.
When each number of successes X is paired with its probability of occurrence Af, the set of pairs Af, is a probability function called a binomial distribution.
The function f{t} defined in this way is multi-valued.
In some neighborhood in the f-plane of any ordinary point of the graph, the function f is a single-valued, continuous function.
is and f
The graph of f has at least one component whose support is the entire interval Aj.
" f one regards the Modest Proposal simply as a criticism of condition, about all one can say is that conditions were bad and that Swift's irony brilliantly underscored this fact ".
A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:
: For any set A there is a function f such that for any non-empty subset B of A, f ( B ) lies in B.
: There is a set A such that for all functions f ( on the set of non-empty subsets of A ), there is a B such that f ( B ) does not lie in B.
* There exists a model of ZF ¬ C in which there is a function f from the real numbers to the real numbers such that f is not continuous at a, but f is sequentially continuous at a, i. e., for any sequence
There is one final criterion that Hume thinks gives us warrant to doubt any given testimony, and that is f ) if the propositions being communicated are miraculous.
* An example of a non-unitary associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.
is and has
`` And if the dive goes OK he has the exclusive import rights to your line for this country, is that right ''??
In fact it has caused us to give serious thought to moving our residence south, because it is not easy for the most objective Southerner to sit calmly by when his host is telling a roomful of people that the only way to deal with Southerners who oppose integration is to send in troops and shoot the bastards down.
As it is, they consider that the North is now reaping the fruits of excess egalitarianism, that in spite of its high standard of living the `` American way '' has been proved inferior to the English and Scandinavian ways, although they disapprove of the socialistic features of the latter.
Historically, however, the concept is one that has been of marked benefit to the people of the Western civilizational group.
While the pattern is uneven, some having gained more than others, nationalism has in fact served the Western peoples well.
But it is more than irony: one of the main reasons why nationalism is no longer a tenable concept is because it has spread throughout the planet.
While sovereignty has roots in antiquity, in its present usage it is essentially modern.
It has nothing of the proud stride of the trained runner about it, it is not a lope, it is not done with style or verve.
It is softened by the saltbush and the bluebush, has a peaceful quality, the hills roll softly.
The music director of the Pittsburgh Symphony Orchestra, William Steinberg, has molded his group into a prominent musical organization, which is his life.
`` Now that Bruno Walter is virtually in retirement and my dear friend Dimitri Mitropoulos is no longer with us, I am probably the only one -- with the possible exception of Leonard Bernstein -- who has this special affinity for and champions the works of Bruckner and Mahler ''.
Even though in most cases the completion of the definitive editions of their writings is still years off, enough documentation has already been assembled to warrant drawing a new composite profile of the leadership which performed the heroic dual feats of winning American independence and founding a new nation.
It is interesting, however, that despite this strong upsurge in Southern writing, almost none of the writers has forsaken the firmly entrenched concept of the white-suited big-daddy colonel sipping a mint julep as he silently recounts the revenue from the season's cotton and tobacco crops ; ;
But the South is, and has been for the past century, engaged in a wide-sweeping urbanization which, oddly enough, is not reflected in its literature.
In the meantime, while the South has been undergoing this phenomenal modernization that is so disappointing to the curious Yankee, Southern writers have certainly done little to reflect and promote their region's progress.
An approach that has appealed to some choreographers is reminiscent of Charles Olson's statement of the process of projective verse: `` one perception must immediately and directly lead to a further perception ''.
The process stipulates that the choreographer sense the quality of the initial movement he has discovered and that he feel the rightness of the quality that is to follow it.
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