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Page "Cauchy–Riemann equations" ¶ 70
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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 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 >.
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 >.
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
If ƒ ( x )
* 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 z
If we are to keep a right-handed coordinate system, into the page will be in the negative z direction.
If the axes are named x, y, and z, then the x coordinate is the distance from the plane defined by the y and z axes.
If the measurement result is + z, this means that immediately after measurement the system state undergoes an orthogonal projection of ψ onto the
* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if z is an element of V not in the closure of U, then there exists a continuous linear map with ψ ( x ) = 0 for all x in U, ψ ( z ) = 1, and || ψ || = 1 / dist ( z, U ).
If two quarks have unaligned spins, the spin vectors add up to make a vector of length S = 0 and only one spin projection ( S < sub > z </ sub > = 0 ), called the spin-0 singlet.
Alice could send bits to Bob in the following way: If Alice wishes to transmit a " 0 ", she measures the spin of her electron in the z direction, collapsing Bob's state to either or.
* If x < y and y < z, then x < z.
* If x < y and z > 0, then xz < yz.
If λ represents wavelength and f represents frequency ( note, λf = c where c is the speed of light ), then z is defined by the equations:
If an electron of energy E is incident upon an energy barrier of height U ( z ), the electron wave function is a traveling wave solution,
:: If Alice wishes to transmit a ' 0 ', she measures the spin of her electron in the z direction, collapsing Bob's state to either | z +>< sub > B </ sub > or | z ->< sub > B </ sub >.
If the right-hand side is specified as a given function, f ( x, y, z ), i. e., if the whole equation is written as
If z is the vector, one has
If z is a complex number, written in polar form as

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.

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