In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P. It states that such a function is, up to multiplication by a function not zero at P, a polynomial in one fixed variable z, which is monic, and whose coefficients are analytic functions in the remaining variables and zero at P.

This class of functions are also referred to as P-functions and generally written using the symbol &# 8472 ; ( or ), and known as " Weierstrass P ").

Symbol for Weierstrass P function < div class =" thumbcaption ">

Symbol for Weierstrass P function

Weierstrass P function defined over a subset of the complex plane using a standard visualization technique in which white corresponds to a pole, black to a zero, and maximal saturation to Note the regular lattice of poles, and two interleaving lattices of zeros.

* Mathematics, the Dirichlet eta function, Dedekind eta function, and Weierstrass eta function.

The Weierstrass factorization theorem asserts that any entire function can be represented by a product involving its zeroes.

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.

Thus, it was not until two centuries had passed that in 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable.

In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function.

Further, there is a generalization of the Stone – Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space is approximated uniformly on compact sets by algebras of the type appearing in the Stone – Weierstrass theorem and described below.

As a consequence of the Weierstrass approximation theorem, one can show that the space C is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients ; there are only countably many polynomials with rational coefficients.

* ln ( a ) if a is algebraic and not equal to 0 or 1, for any branch of the logarithm function ( by the Lindemann – Weierstrass theorem ).

* Weierstrass zeta function This is related to elliptic functions and is not analogous to the Riemann zeta function.

A more complete study of elliptic functions was later undertaken by Karl Weierstrass, who found a simple elliptic function in terms of which all the others could be expressed.

Although Jacobi's elliptic functions are older and more directly relevant to applications, modern authors mostly follow Karl Weierstrass when presenting the elementary theory, because his functions are simpler, and any elliptic function can be expressed in terms of them.

With the definition of elliptic functions given above ( which is due to Weierstrass ) the Weierstrass elliptic function is constructed in the most obvious way: given a lattice as above, put

The formal definition of continuity of a function, as formulated by Weierstrass, is as follows:

There are certainly examples that are neither one nor the other — these occur for example in the modern theory of integrals of the second and third kinds such as the Weierstrass zeta function, or the theory of generalized Jacobians.

In analysis, Kronecker rejected the formulation of a continuous, nowhere differentiable function by his colleague, Karl Weierstrass.

The Weierstrass function is continuous, but is not differentiable at any point.

The first known example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function.

* Weierstrass sigma function, related to elliptic functions

* e < sup > a </ sup > if a is algebraic and nonzero ( by the Lindemann – Weierstrass theorem ).

In mathematics, specifically in real analysis, the Bolzano – Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space R < sup > n </ sup >.

gh + j, where j is a polynomial of degree less than N. This is equivalent to the preparation theorem, since the Weierstrass factorization of f may be obtained by applying the division theorem for g = z < sup > N </ sup > for the least N that gives an h not zero at the origin ; the desired Weierstrass polynomial is then z < sup > N </ sup > + j / h.

There is an analogous result, also referred to as the Weierstrass preparation theorem, for power series rings over the ring of integers in a p-adic field ; namely, a power series f ( z ) can always be uniquely factored as π < sup > n </ sup >· u ( z )· p ( z ), where u ( z ) is a unit in the ring of power series, p ( z ) is a distinguished polynomial ( monic, with the coefficients of the non-leading term each in the maximal ideal ), and π is a fixed uniformizer.

* The Bolzano – Weierstrass theorem, which ensures compactness of closed and bounded sets in R < sup > n </ sup >

Notationally, the quarter periods K and iK ′ are usually used only in the context of the Jacobian elliptic functions, whereas the half-periods ω < sub > 1 </ sub > and ω < sub > 2 </ sub > are usually used only in the context of Weierstrass elliptic functions.

In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g < sub > 2 </ sub > = 0 and g < sub > 3 </ sub > = 1.

The special case where the numbers z < sub > 1 </ sub >,..., z < sub > n </ sub > are all algebraic is the Lindemann – Weierstrass theorem.

During the 1890s Bukreev published a series of high quality papers including: " On the theory of gamma functions ," " On some formulas in the theory of elliptic functions of Weierstrass ," " On the distribution of the roots of a class of entire transcendental functions ," and " Theorems for elliptic functions of Weierstrass ".

Indeed, let K be an imaginary quadratic field with class field H. Let E be an elliptic curve with complex multiplication by the integers of K, defined over H. Then the maximal abelian extension of K is generated by the x-coordinates of the points of finite order on some Weierstrass model for E over H.

Indeed, by the Stone – Weierstrass theorem, any continuous function on the unit interval can be expressed as a limit of polynomials, so any class of functions containing the polynomials and closed under limits will necessarily include all continuous functions.

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.

In general, P is a function of V and the current density, but it shall here be assumed as a constant.

An optimal policy is an admissible policy Af which maximizes the objective function P.

The policy may not be unique but the maximum value of P certainly is, and once the policy is specified this maximum can be calculated by ( 2 ) and ( 3 ) as a function of the feed state Af.

For each K, the function E < sub > K </ sub >( P ) is required to be an invertible mapping on

A quadratic Bézier curve is the path traced by the function B ( t ), given points P < sub > 0 </ sub >, P < sub > 1 </ sub >, and P < sub > 2 </ sub >,

Let P < sub > F </ sub > be the domain of a prefix-free universal computable function F. The constant Ω < sub > F </ sub > is then defined as

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant ( i. e., elements of their transfer function C ( s ), P ( s ), and F ( s ) do not depend on time ), the systems above can be analysed using the Laplace transform on the variables.

Then, using the periodic Bernoulli function P < sub > n </ sub > defined above and repeating the argument on the interval, one can obtain an expression of ƒ ( 1 ).

Power sets: The power set functor P: Set → Set maps each set to its power set and each function to the map which sends to its image.

The graphs on the right side depict the ( finite ) coefficients that modulate the infinite amplitudes of a comb function whose teeth are spaced at intervals of 1 / P.

The Fourier transform of a periodic function, s < sub > P </ sub >( t ), with period P, becomes a Dirac comb function, modulated by a sequence of complex coefficients:

When s < sub > P </ sub >( t ), is expressed as a periodic summation of another function, s ( t ):

Consider a pseudo random number generator ( PRNG ) function P ( key ) that is uniform on the interval 2 < sup > b </ sup > − 1.

A hash function uniform on the interval n-1 is n P ( key )/ 2 < sup > b </ sup >.

According to K. F. P. v. Martius the kanaima is a human being who employs poison to carry out his function of blood avenger ; other authorities represent the kanaima as a jaguar, which is either an avenger of blood or the familiar of a cannibalistic sorcerer.

For example assume that the monopoly ’ s demand function is P

The Curry-Howard correspondence between proofs and programs relates modus ponens to function application: if f is a function of type P → Q and x is of type P, then f x is of type Q.