 Page "Holomorphic function" ¶ 7
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If and ƒ If ƒ is differentiable, this is equivalent to: 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 ). * 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. If X is a set and M is a complete metric space, then the set B ( X, M ) of all bounded functions ƒ from X to M is a complete metric space. If X is a topological space and M is a complete metric space, then the set C < sub > b </ sub >( X, M ) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B ( X, M ) and hence also complete. If and are groups, a homomorphism from to is a function ƒ: → such that If the limit exists, we say that ƒ is complex-differentiable at the point z < sub > 0 </ sub >. In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. If ƒ maps X to Y, then ƒ < sup >– 1 </ sup > maps Y back to X. If ƒ is invertible, the function g is unique ; in other words, there can be at most one function g satisfying this property. If the domain is the real numbers, then each element in Y would correspond to two different elements in X (± x ), and therefore ƒ would not be invertible. If ƒ is an invertible function with domain X and range Y, then If an inverse function exists for a given function ƒ, it is unique: it must be the inverse relation. If ƒ is real-valued, the polynomial function can be taken over R. If D is a derivation at x, then D ( ƒ ) If ƒ and g are elements of a C *- algebra, f * and g * denote their respective adjoints. If ƒ: A → B is a homomorphism between two algebraic structures ( such as homomorphism of groups, or a linear map between vector spaces ), then the relation ≡ defined by * Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.

If and is If the circumstances are faced frankly it is not reasonable to expect this to be true. If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities. If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance. If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ; If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz. If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil. If it is an honest feeling, then why should she not yield to it?? If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control. If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre. If he is good, he may not be legal ; ; If the man on the sidewalk is surprised at this question, it has served as an exclamation. If the existent form is to be retained new factors that reinforce it must be introduced into the situation. If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis?? If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego. If he is a traditionalist, he is an eclectic traditionalist. If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination. If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If and complex If we are discussing differentiable complex-valued functions, then Af and V are complex vector spaces, and Af may be any complex numbers. If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K. If A is a self-adjoint operator, then is always a real number ( not complex ). If more than one eigenvalue is negative, then the stationary point is a more complex one, and is usually of little interest. If X is Cauchy distributed with median μ and scale parameter γ, then the complex variable If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable: in this case the system transfer function has non-repeated poles at complex plane origin ( i. e. their real and complex component is zero in the continuous time case ). If the weight of the cable and supporting wires are not negligible then the analysis is more complex. If a stitch in a finished crocheted item breaks, the stitches above and below remain intact, and because of the complex looping of each stitch, the stitches on either side are unlikely to come loose unless heavily stressed. If we look for solutions that have the form, where C is a constant, we discover the relationship, and thus must be one of the complex numbers or. If < math > c ^ 2 < 4km </ math > there are two complex conjugate roots a ± ib, and the solution ( with the above boundary conditions ) will look like this: If anything, the research has shown the situation is even more complex than initially imagined, and we believe not enough is known to structure the classification of psychiatric disorders according to etiology. If the obstructing object provides multiple, closely spaced openings, a complex pattern of varying intensity can result. If the ether is part of a more complex molecule, it is described as an alkoxy substituent, so-OCH < sub > 3 </ sub > would be considered a " methoxy -" group. If the cell is synthesizing DNA each pore complex needs to transport about 100 histone molecules per minute. If the cell is growing rapidly, each complex also needs to transport about 6 newly assembled large and small ribosomal subunits per minute from the nucleus to the cytosol, where they are used to synthesize proteins. If it is unaffected, then it is a simple partial seizure ; otherwise it is a complex partial ( psychomotor ) seizure. 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: If the coordinates are chosen badly, the laws of motion may be more complex than necessary. In The Origin of Species, he wrote, " If it could be demonstrated that any complex organ existed, which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down. If ( i ) or ( j ) have side effects or introduce run time errors, the subtle differences between ( l ) and ( l ') can be complex. If, however, the set of allowed candidates is expanded to the complex numbers, every non-constant polynomial has at least one root ; this is the fundamental theorem of algebra. * If R denotes the ring CY of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y < sup > 2 </ sup > − X < sup > 3 </ sup > − X − 1 is a prime ideal ( see elliptic curve ).

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