Suppose we are interested in the modes of vibration of a thin membrane, held in a rectangular frame, 0 < x < L1, 0 < y < L2. We know the equation of motion for the vertical membrane's displacement, W(x, y, t) is given by the wave equation: The equation is separable (substituting W = X(x) × Y(y) × T(t)), and the normal mode solutions that have harmonic time dependence and sati ...

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Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville form, Sturm-Liouville theory - Examples, Sturm-Liouville theory - Sturm-Liouville differential operators, Sturm-Liouville theory - Some highly technical details, Sturm-Liouville theory - Useful consequences of the preceding technicalities, Sturm-Liouville theory - Example, Sturm-Liouville theory - Application to normal modes

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It is natural to ask, for which classes of functions and for which interpolation nodes the sequence of interpolating polynomials converges to the interpolated function? Convergence may be understood in different ways, e.g. pointwise, uniform or in some integral norm. The aspects of uniform convergence are discussed below. The following theorem seems to be a rather encouraging answer: For any function f(x) continuous on an interval [a,b] there exists a table of nodes for which the sequenc ...

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Polynomial interpolation, Polynomial interpolation - Applications, Polynomial interpolation - Definition, Polynomial interpolation - Constructing the interpolation polynomial, Polynomial interpolation - Non-Vandermonde solutions, Polynomial interpolation - Interpolation error, Polynomial interpolation - Lebesgue constants, Polynomial interpolation - Convergence properties, Polynomial interpolation - Related concepts

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The empty set is a set of uniqueness. This is just a fancy way to say that if a trigonometric series converges to zero everywhere then it is trivial. This was proved by Riemann, using a delicate technique of double formal integration; and showing that the resulting sum has some generalized kind of second derivative using Toeplitz operators. Later on, Cantor generalized Riemann's techniques to show that any countable, closed set is a set of uniqueness, a disco ...

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Set of uniqueness, Set of uniqueness - Definition, Set of uniqueness - Early research, Set of uniqueness - Transformations, Set of uniqueness - Singular distributions, Set of uniqueness - Complexity of structure

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A hypergeometric series could in principle be any formal power series in which the ratio of successive summands is a rational function of n. That is, for some polynomials and . Thus, for example, in the case of a geometric series, this ratio is a constant. Another example is the series for the exponential function, for which In practice the series is written as an exponential generating function, modifying the coefficients so that a general ...

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Hypergeometric series, Hypergeometric series - Introduction, Hypergeometric series - Notation, Hypergeometric series - Special cases and applications, Hypergeometric series - Identities, Hypergeometric series - History and generalizations

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In a formula: the geometric mean of a1, a2, ..., an is , which is . The geometric mean of a data set is always smaller than or equal to the set's arithmetic mean (the two means are equal if and only if all members of the data set are equal). This allows the definition of the arithmetic-geometric mean, a mixture of the two which always lies in between. The geometric mean is also the arithmetic-harmonic mean in the sense that if two sequences (an) and (hn) are defined: See also:

Geometric mean, Geometric mean - Calculation, Geometric mean - Relationship with arithmetic mean of logarithms, Geometric mean - When to use the Geometric Mean

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It is a standard theorem of measure theory that the completion of Cc(G) in the L1(G) norm is isomorphic to the space L1(G) of functions which are integrable with respect to the Haar measure. Theorem. L1(G) is a B*-algebra with the convolution product and involution defined above and with the L< ...

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Group algebra, Group algebra - Group algebra of a finite group, Group algebra - Group algebras of topological groups: CcG, Group algebra - The convolution algebra L1G, Group algebra - The group C*-algebra C*G, Group algebra - The maximal group C * -algebra, Group algebra - The reduced group C*-algebra C*rG, Group algebra - von Neumann algebras associated to groups

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Such a theory cannot exist in the same form for non-commutative groups G, since in that case the appropriate dual object G^ of isomorphism classes of representations cannot only contain one-dimensional representations, and will fail to be a group. The generalisation that has been found useful in category theory is called Tannaka-Krein duality; but this diverges from the connection with harmonic analysis, which needs to tackle the question of the Plancherel measure on G^. There are analogues of duality theory for noncommutative groups, ...

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Pontryagin duality, Pontryagin duality - Haar measure, Pontryagin duality - The dual group, Pontryagin duality - Fourier transform, Pontryagin duality - Examples, Pontryagin duality - The group algebra, Pontryagin duality - Plancherel and Fourier inversion theorems, Pontryagin duality - Bohr compactification and almost-periodicity, Pontryagin duality - Categorical considerations, Pontryagin duality - Non-commutative theory, Pontryagin duality - History

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The inverse Laplace transform is the Bromwich integral, which is a complex integral given by: where is a real number so that the contour path of integration is in the region of convergence of normally requiring for every singularity of and . If all singularities are in the left half-plane, that is for every , then can be set to zero and the above inverse integral fo ...

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Laplace transform, Laplace transform - Formal definition, Laplace transform - Region of convergence, Laplace transform - Inverse Laplace transform, Laplace transform - Bilateral Laplace transform, Laplace transform - Laplace transform of a function's derivative, Laplace transform - Applications, Laplace transform - Example #1: Solving a differential equation, Laplace transform - Example #2: Deriving the complex impedance for a capacitor, Laplace transform - Example #3: Finding the transfer function from the impulse response, Laplace transform - Relationship to other transforms, Laplace transform - Fourier transform, Laplace transform - Mellin transform, Laplace transform - Z-transform, Laplace transform - Fundamental relationships, Laplace transform - Properties and theorems, Laplace transform - Table of selected Laplace transforms, Laplace transform - Examples: How to apply the properties and theorems, Laplace transform - Example #1: Method of partial fraction expansion, Laplace transform - Example #2: Mixing sines cosines and exponentials, Laplace transform - Example #3, Laplace transform - Example #4: Phase delay

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The concept of Lp space can be extended to vectors having an infinite number of components. For an infinite sequence of real (or complex) numbers, define the p-norm Here, a complication arises, that being that the series on the right is not always convergent, so for example, the sequence made up of only ones, will have an infinite p-norm (length), no matter what p≥1 is. The space is then defined as the set of all infinite sequences of real numbers such that the p-norm is finite. One can check that as p increases, the se ...

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Lp space, Lp space - Motivation, Lp space - lp spaces, Lp space - Properties of lp spaces, Lp space - Lp spaces, Lp space - Special cases, Lp space - Relation to lp spaces, Lp space - Properties of Lp spaces

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The de Rham cohomology has inspired many mathematical ideas, including Dolbeault cohomology, Hodge theory, and the Atiyah-Singer index theorem. However, even in more classical contexts, the theorem has inspired a number of developments. Firstly, the Hodge theorem proves that there is an isomorphism between the cohomology consisting of harmonic forms and the de Rham cohomology consisting of closed forms modulo exact forms. This relies on an appropriate definition of harmonic forms and of the Hodge theorem. For further details see Hodge theory. < ...

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De Rham cohomology, De Rham cohomology - Definition, De Rham cohomology - de Rham cohomology computed, De Rham cohomology - de Rham's theorem, De Rham cohomology - Sheaf-theoretic de Rham isomorphism, De Rham cohomology - Proof, De Rham cohomology - Related ideas, De Rham cohomology - Harmonic forms, De Rham cohomology - Hodge decomposition

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If this series converges for every x in the interval (a − r, a + r) and the sum is equal to f(x), then the function f(x) is called analytic. To check whether the series converges towards f(x), one normally uses estimates for the remainder term of Taylor's theorem. A function is analytic if and only if it can be represented as a power series; the coefficients in that power series are then necessarily the on ...

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Taylor series, Taylor series - History, Taylor series - Properties, Taylor series - Taylor series for several variables, Taylor series - List of Taylor series of some common functions, Taylor series - Calculation of Taylor series

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If G is a locally compact abelian group, a character of G is a continuous group homomorphism from G with values in the circle group T. It can be shown that the set of all characters on G is itself a locally compact abelian group, called the dual group of G. The group operation on the dual group is given by pointwise multiplication of characters, the inverse of a character is its complex conjugate and the topology on the space of characters is that of uniform convergence on compact sets ...

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Pontryagin duality, Pontryagin duality - Haar measure, Pontryagin duality - The dual group, Pontryagin duality - Fourier transform, Pontryagin duality - Examples, Pontryagin duality - The group algebra, Pontryagin duality - Plancherel and Fourier inversion theorems, Pontryagin duality - Bohr compactification and almost-periodicity, Pontryagin duality - Categorical considerations, Pontryagin duality - Non-commutative theory, Pontryagin duality - History

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For the purposes of functional analysis, and in particular of harmonic analysis, one wishes to carry over the group ring construction to topological groups G. In case G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called Haar measure. Using the Haar measure, one can define a convolution operation on the space Cc(G) of complex-valued functions on G with compact support; C ...

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Group algebra, Group algebra - Group algebra of a finite group, Group algebra - Group algebras of topological groups: CcG, Group algebra - The convolution algebra L1G, Group algebra - The group C*-algebra C*G, Group algebra - The maximal group C * -algebra, Group algebra - The reduced group C*-algebra C*rG, Group algebra - von Neumann algebras associated to groups

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The term "potential theory" arises from the fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from potentials which satisfied Laplace's equation. Hence, potential theory was the study of functions which could serve as potentials. Nowadays, we know that nature is more complicated -- the equations which describe forces are systems of non-linear partial differential equations such as the Einstein equations and the Yang-Mills equations and that the Laplace equation is only valid as a limiting case. ...

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Potential theory, Potential theory - Definition and comments, Potential theory - Symmetry, Potential theory - Two dimensions, Potential theory - Local behavior, Potential theory - Inequalities, Potential theory - Spaces of harmonic functions

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The n-th raw moment is defined as the expected value of kn: The series on the right is just a series representation of the Riemann zeta function, but it only converges for values of s-n that are greater than unity. Thus: Note that the ratio of the zeta functions is well defined, even for because the series representation of the zeta function can be analytically continued. This does not change the fact that the moments are specified by the series itself, an ...

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Zeta distribution, Zeta distribution - Moments, Zeta distribution - Moment generating function, Zeta distribution - The case s = 1

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To obtain the numerical value of the exponential function. The infinite series can be rewritten as : This expression will converge quickly if we can ensure that x is less than one. To ensure this, we can use the following identity. Where z is the integer part of x Where f< ...

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Exponential function, Exponential function - Properties, Exponential function - Derivatives and differential equations, Exponential function - Formal definition, Exponential function - Numerical value, Exponential function - On the complex plane, Exponential function - Matrices and Banach algebras, Exponential function - On Lie algebras, Exponential function - Double exponential function

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The Taylor series may also be generalised to functions of more than one variable with A second-order Taylor series expansion of a scalar-valued function of more than one variable can be compactly written as where is the gradient and is the Hessian matrix. Applying the multi-index notation the Taylor series for several variables becomes ...

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Taylor series, Taylor series - History, Taylor series - Properties, Taylor series - Taylor series for several variables, Taylor series - List of Taylor series of some common functions, Taylor series - Calculation of Taylor series

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Several important Taylor/Maclaurin series expansions follow. All these expansions are also valid for complex arguments x. Exponential function and natural logarithm: Geometric series: Binomial theorem: Trigonometric functions: Hyperbolic functions: See also:

Taylor series, Taylor series - History, Taylor series - Properties, Taylor series - Taylor series for several variables, Taylor series - List of Taylor series of some common functions, Taylor series - Calculation of Taylor series

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A closed set is a set of uniqueness if and only if there exists a distribution S supported on the set (so in particular it must be singular) such that ( here are the Fourier coefficients). In all early examples of sets of uniqueness the distribution in question was in fact a measure. In 1954, though, Ilya Pyatetskii-Shapiro constructed an example of a set of uniqueness which does not support any measure with Fourier coefficients tending to zero. In o ...

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Set of uniqueness, Set of uniqueness - Definition, Set of uniqueness - Early research, Set of uniqueness - Transformations, Set of uniqueness - Singular distributions, Set of uniqueness - Complexity of structure

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Suppose we have a current loop with a current I in it. Then the vector potential of the induced magnetic field is and as before we can expand in negative powers of , obtaining another multipole expansion: The n = 0 term is always zero, since it equals the integral of a constant function around a closed loop. (This term, if present, would describe magnetic monopoles; if those existed, there would be no such thing as a magnetic field's vector potential.) The n = 1 term is the dipole ...

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Multipole expansion, Multipole expansion - Multipole expansion for electric potentials, Multipole expansion - Multipole expansion for electric fields, Multipole expansion - Multipole expansion for magnetic vector potentials, Multipole expansion - Multipole expansion for magnetic fields, Multipole expansion - Multipole expansions in electrodynamics, Multipole expansion - Applications of the multipole expansion

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Let us now consider the potential V(r) = V0 for r < r0, i.e., inside a sphere of radius r0 and zero outside. We first consider bound states, i.e., states which display the particle mostly inside the box (confined states). Those have an energy E less than the potential outside the sphere, i.e., they have negative energy, and we shall see that there are a discrete num ...

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Particle in a spherically symmetric potential, Particle in a spherically symmetric potential - General considerations, Particle in a spherically symmetric potential - Vacuum case, Particle in a spherically symmetric potential - Spherical square well, Particle in a spherically symmetric potential - Infinite spherical square well

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The dual group of a locally compact abelian group is introduced as the underlying space for an abstract version of the Fourier transform. If a function is in L1(G), then the Fourier transform is the function on G^ such that where the integral is relative to Haar measure μ on G. It is not too difficult to show that the Fourier transform of an L1 function on G is a bounded continuous function on G^ which vanishes at infinity. Similarly, the inverse Fourier transform of an integrabl ...

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Pontryagin duality, Pontryagin duality - Haar measure, Pontryagin duality - The dual group, Pontryagin duality - Fourier transform, Pontryagin duality - Examples, Pontryagin duality - The group algebra, Pontryagin duality - Plancherel and Fourier inversion theorems, Pontryagin duality - Bohr compactification and almost-periodicity, Pontryagin duality - Categorical considerations, Pontryagin duality - Non-commutative theory, Pontryagin duality - History

Read more here: » Pontryagin duality: Encyclopedia II - Pontryagin duality - Fourier transform