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Second-order linear differential equation
Chebyshev's equation is the second order linear differential equation ( 1 − x 2 ) d 2 y d x 2 − x d y d x + p 2 y = 0 , {\displaystyle (1-x^{2}){d^{2}y
Chebyshev_equation
Type of analog or digital filter
Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband
Chebyshev_filter
Bound on probability of a random variable being far from its mean
In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) provides an upper bound on the probability of deviation of
Chebyshev's_inequality
Russian mathematician (1821–1894)
roots of equations" which he had finished in 1838. In this, Chebyshev derived an approximating algorithm for the solution of algebraic equations of nth
Pafnuty_Chebyshev
Pair of polynomial sequences
The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)}
Chebyshev_polynomials
Chebyshev center Chebyshev constants Chebyshev cube root Chebyshev distance Chebyshev equation Chebyshev's equioscillation theorem Chebyshev filter, a
List of things named after Pafnuty Chebyshev
List_of_things_named_after_Pafnuty_Chebyshev
Polynomial equation of degree 3
the proper Chebyshev cube root. The value involving hyperbolic sines is similarly denoted S1/3(q), when p = 3. For solving the cubic equation x3 + m2x =
Cubic_equation
Type of Diophantine equation
Pell's equation and the Chebyshev polynomials: If T i ( x ) {\displaystyle T_{i}(x)} and U i ( x ) {\displaystyle U_{i}(x)} are the Chebyshev polynomials
Pell's_equation
Mathematical mentod
ISBN 978-0-486-61272-0. Equation 25.4.40. Chebyshev-Gauss Quadrature from Wolfram MathWorld Gauss–Chebyshev type 1 quadrature and Gauss–Chebyshev type 2 quadrature
Chebyshev–Gauss_quadrature
Mathematical function
mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function ϑ(x)
Chebyshev_function
Orbital mechanics term
for a transcendental equation without a first guess: Polynomialization of Kepler's equation through Chebyshev polynomial equation of the sine". Applied
Kepler's_equation
Theory of getting acceptably close inexact mathematical calculations
function, using the Chebyshev polynomials instead of the usual trigonometric functions. If one calculates the coefficients in the Chebyshev expansion for a
Approximation_theory
Four-bar straight-line mechanism
In kinematics, Chebyshev's linkage is a four-bar linkage that converts rotational motion to approximate linear motion. It was invented by the 19th-century
Chebyshev_linkage
numerical linear algebra, the Chebyshev iteration is an iterative method for determining the solutions of a system of linear equations. The method is named after
Chebyshev_iteration
Polynomial sequence
α = 1, the equation reduces to the Chebyshev differential equation, and the Gegenbauer polynomials reduce to the Chebyshev polynomials of the second kind
Gegenbauer_polynomials
This is a list of scientific equations named after people (eponymous equations). Contents A B C D E F G H I J K L M N O P R S T V W Y Z See also References
List of scientific equations named after people
List_of_scientific_equations_named_after_people
and number theory, author of the Chebyshev's inequality, Chebyshev distance, Chebyshev function, Chebyshev equation etc. Sergei Chernikov, significant
List of Russian mathematicians
List_of_Russian_mathematicians
Böttcher's equation, named after Lucjan Böttcher, is the functional equation F ( h ( z ) ) = ( F ( z ) ) n {\displaystyle F(h(z))=(F(z))^{n}} where h
Böttcher's_equation
Polynomial sequence
scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles
Hermite_polynomials
{E} _{0}^{+}&={\text{Non-negative even integers}}\\\end{aligned}}} The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula
List of trigonometric identities
List_of_trigonometric_identities
and number theory, author of the Chebyshev's inequality, Chebyshev distance, Chebyshev function, Chebyshev equation Boris Delaunay, inventor of Delaunay
List_of_Russian_scientists
Russian mathematician (1857–1918)
equations. He created the modern theory of the stability of a dynamical system. In the theory of probability, he generalized the works of Chebyshev and
Aleksandr_Lyapunov
Type of orthogonal polynomials
polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials). They have many important applications
Classical orthogonal polynomials
Classical_orthogonal_polynomials
Transport of a substance by bulk motion
confused with the notation for partial derivatives. Boyd, John P. (2001). Chebyshev and Fourier Spectral Methods (PDF). Mineola, NY: Courier Corporation.
Advection
Function defined by a hypergeometric series
ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. For systematic
Hypergeometric_function
Numerical technique
1 < t 1 < … < t p < 1 {\displaystyle -1<t_{1}<\ldots <t_{p}<1} be the Chebyshev nodes of order p ≥ 2 , {\displaystyle p\geq 2,} and let u 1 ( y ) , …
Fast_multipole_method
Equation whose side(s) describe a transcendental function
1993) [in Russian] John P. Boyd (2014). Solving Transcendental Equations: The Chebyshev Polynomial Proxy and Other Numerical Rootfinders, Perturbation
Transcendental_equation
Simple curve of Euclidean geometry
for the Chebyshev distance (L∞ metric) on a plane is also a square with side length 2r parallel to the coordinate axes, so planar Chebyshev distance
Circle
Frequency response boundary
ratios besides the 3 dB point may also be relevant, for example see § Chebyshev filters below. Far from the cutoff frequency in the transition band, the
Cutoff_frequency
Signal processing method
ripples by solving a set of nonlinear equations. Another method introduced at the time implemented an optimal Chebyshev approximation, but the algorithm was
Parks–McClellan filter design algorithm
Parks–McClellan_filter_design_algorithm
fractional Chebyshev collocation (FCC) method is an efficient spectral method for solving a system of linear fractional-order differential equations (FDEs)
Fractional Chebyshev collocation method
Fractional_Chebyshev_collocation_method
Algorithm for finding zeros of functions
Taylor approximation. In the 19th century, Russian mathematician Pafnuty Chebyshev explored this idea by developing a variant of Newton’s method that used
Newton's_method
System of complete and orthogonal polynomials
multipole expansion. The trigonometric functions cos nθ, also denoted as the Chebyshev polynomials Tn(cos θ) ≡ cos nθ, can also be multipole expanded by the
Legendre_polynomials
Concept in probability theory
(sometimes, calling it the first Chebyshev inequality, while referring to Chebyshev's inequality as the second Chebyshev inequality) or Bienaymé's inequality
Markov's_inequality
and number theory, author of the Chebyshev's inequality, Chebyshev distance, Chebyshev function, Chebyshev equation Boris Delaunay, inventor of Delaunay
List_of_Russian_people
Type of metric geometry
for the Chebyshev distance (L∞ metric) on a plane is also a square with side length 2r parallel to the coordinate axes, so planar Chebyshev distance
Taxicab_geometry
Numerical analysis technique for waves
derivative terms of the wave equation are evaluated in the spectral domain using orthogonal bases, such as Fourier or Chebyshev bases while the problem is
Pseudospectral time-domain method
Pseudospectral_time-domain_method
Technique in mathematics
technique which replaces a nonlinear differential equation or operator equation (or system of such equations) with a sequence of linear problems, which are
Quasilinearization
Complex exponential in terms of sine and cosine
(in radians). Considering cos x a parameter in equation above yields recursive formula for Chebyshev polynomials of the first kind. In the language of
Euler's_formula
Analytic function in mathematics
the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to Re(s) > 1. The above series is a prototypical
Riemann_zeta_function
Algorithm to approximate functions
usually the extrema of Chebyshev polynomial linearly mapped to the interval. The steps are: Solve the linear system of equations b 0 + b 1 x i + . . .
Remez_algorithm
Signal processing filter
it is possible to convert a Chebyshev filter characteristic equation, K ( s ) {\displaystyle K(s)} containing the Chebyshev reflection zeros in the numerator
Elliptic_filter
Russian mathematician (1856–1922)
calculus), Pafnuty Chebyshev (number theory and probability theory), Aleksandr Korkin (ordinary and partial differential equations), Mikhail Okatov (mechanism
Andrey_Markov
Method for solving differential equations
residuals (MWR) are methods for solving differential equations. The solutions of these differential equations are assumed to be well approximated by a finite
Method of mean weighted residuals
Method_of_mean_weighted_residuals
Class of methods used in numerical analysis and scientific computing to solve ODE/PDE
numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis functions"
Spectral_method
Result of repeatedly applying a mathematical function
functional equation, cf. Schröder's equation and Abel equation. On a logarithmic scale, this reduces to the nesting property of Chebyshev polynomials
Iterated_function
Type of signal filter
the horizontal line. The various types of filters (Butterworth filter, Chebyshev filter, Bessel filter, etc.) all have different-looking knee curves. Many
Low-pass_filter
Sequence of mathematical functions
mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal. They are named after Pafnuty Chebyshev. A rational
Chebyshev_rational_functions
Function specifying the behavior of a component in an electronic or control system
given order Chebyshev filter (Type I) – maximally flat in stopband, sharper cutoff than a Butterworth filter of the same order Chebyshev filter (Type
Transfer_function
Function that, applied twice, gives another function
functional square root of g(x) = 8x4. A functional square root of the nth Chebyshev polynomial, g ( x ) = T n ( x ) {\displaystyle g(x)=T_{n}(x)} , is f (
Functional_square_root
type Brahmagupta polynomials Caloric polynomial Charlier polynomials Chebyshev polynomials Chihara–Ismail polynomials Cyclotomic polynomials Dickson
List_of_polynomial_topics
Device for suppressing part of a discretely-sampled signal
nonlinear dynamics. Bessel filter Bilinear transform Butterworth filter Chebyshev filter Electronic filter Elliptical filter (Cauer filter) Filter design
Digital_filter
Type of function
called Legendre rational functions and Chebyshev rational functions. Solutions of linear differential equations with boundary conditions can often be written
Orthogonal_functions
Type of signal processing filter
non-monotonic ripple in the passband or the stopband. Compared with a Chebyshev Type I/Type II filter or an elliptic filter, the Butterworth filter has
Butterworth_filter
essentially equivalent to Chebyshev polynomials with a change of variable, and, in fact, Dickson polynomials are sometimes called Chebyshev polynomials. Dickson
Dickson_polynomial
Length of a line segment
Other common distances in real coordinate spaces and function spaces: Chebyshev distance (L∞ distance), which measures distance as the maximum of the
Euclidean_distance
for analysing boundary value problems for linear partial differential equations and for an important class of nonlinear PDEs belonging to the so-called
Fokas_method
limit Order of accuracy — rate at which numerical solution of differential equation converges to exact solution Series acceleration — methods to accelerate
List of numerical analysis topics
List_of_numerical_analysis_topics
unity, it depends on the prime factorization of n. Prime omega functions Chebyshev functions Liouville function: Λ ( n ) = ( − 1 ) Ω ( n ) {\displaystyle
List of mathematical functions
List_of_mathematical_functions
Discontinued online library
matrices. Some of the supported solutions methods are: Richardson Iteration Chebyshev Iteration Conjugate Gradient (CG) Conjugate Gradient Squared (CGS) BiConjugate
IML++
differential equations. List of nonlinear ordinary differential equations List of nonlinear partial differential equations List of named differential equations Vallée
List of linear ordinary differential equations
List_of_linear_ordinary_differential_equations
Image processing technique
the idea of combining the filter with an iterative method, e.g., the Chebyshev iteration and the conjugate gradient method are proposed in for graph-based
Edge-preserving_smoothing
of things named after Arthur Cayley List of things named after Pafnuty Chebyshev List of things named after John Horton Conway List of things named after
Lists_of_mathematics_topics
Conjecture on zeros of the zeta function
showed that the generalized Riemann hypothesis implies a conjecture of Chebyshev that lim x → 1 − ∑ p > 2 ( − 1 ) ( p + 1 ) / 2 x p = + ∞ , {\displaystyle
Riemann_hypothesis
Iterative method used to solve a linear system of equations
Richardson iteration is an iterative method for solving a system of linear equations. Richardson iteration was proposed by Lewis Fry Richardson in work that
Modified_Richardson_iteration
Mathematical description of quantum state
Sturm–Liouville equations in the setting of Hilbert space. These include the Legendre and Laguerre polynomials as well as Chebyshev polynomials, Jacobi
Wave_function
Russian mathematician
contribution to the development of partial differential equations, and was second only to Chebyshev among the founders of the Saint Petersburg Mathematical
Aleksandr_Korkin
for solving differential equations because it can be used to swiftly and efficiently go from grid point values to Chebyshev series coefficients. Generalized
List of Fourier-related transforms
List_of_Fourier-related_transforms
German mathematician (1826–1866)
{\displaystyle \pi (x)} . Riemann knew of Pafnuty Chebyshev's work on the Prime Number Theorem. Chebyshev had visited Dirichlet in 1852. Riemann's works
Bernhard_Riemann
Mathematical transform that expresses a function of time as a function of frequency
heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can be formally defined as an improper Riemann integral
Fourier_transform
Branch of physics
marching-in-time computational techniques for Maxwell's equations uses either discrete Fourier or discrete Chebyshev transforms to calculate the spatial derivatives
Computational electromagnetics
Computational_electromagnetics
Method for solving differential equations
DOPRI method, is an embedded method for solving ordinary differential equations (ODE). The method is a member of the Runge–Kutta family of ODE solvers
Dormand–Prince_method
Russian mathematician (1847–1878)
1864 because he was too young. Among his academic teachers were Somov, Chebyshev and Aleksandr Korkin, with whom he would have a tight scientific friendship
Yegor_Ivanovich_Zolotaryov
of elliptic electronic filters. (These functions are sometimes called Chebyshev rational functions, not to be confused with certain other functions of
Elliptic_rational_functions
Mathematical curve outputted from a specific pair of parametric equations
1 N π 2 {\displaystyle \delta ={\frac {N-1}{N}}{\frac {\pi }{2}}} are Chebyshev polynomials of the first kind of degree N. This property is exploited
Lissajous_curve
Failure of convergence in interpolation
[citation needed] Chebyshev interpolation (i.e., on Chebyshev nodes) converges uniformly for every absolutely continuous function. Chebyshev nodes Compare
Runge's_phenomenon
Root-finding algorithm
Boyd, John P. (2013). "Finding the Zeros of a Univariate Equation: Proxy Rootfinders, Chebyshev Interpolation, and the Companion Matrix". SIAM Review. 55
Halley's_method
Technique in digital signal processing
Goertzel Algorithm by Kevin Banks Analysis of the Goertzel Algorithm by Uwe Beis in which he compares it to analog 2nd order Chebyshev low pass filter
Goertzel_algorithm
Cayley–Capelli operator Celine's polynomial Charlier polynomial Pafnuty Chebyshev: Chebyshev polynomials Elwin Bruno Christoffel, Darboux: Christoffel–Darboux
List of eponyms of special functions
List_of_eponyms_of_special_functions
Eighth letter of the Greek alphabet
A special function ϑ(z; τ) of several complex variables θ. The first Chebyshev function θ(x) in prime number theory The potential temperature in meteorology
Theta
Resistive and inductive circuit
The first equation is solved by using an integrating factor and yields the current which must be differentiated to give VL; the second equation is straightforward
RL_circuit
Filter in electronics and signal processing
and the lower and mid frequencies, but switches to a higher steepness Chebyshev attenuation at the higher frequencies. The Gaussian function is for x
Gaussian_filter
Numerical method for solving optimal control problems
control. Examples of these are the Legendre pseudospectral method, the Chebyshev pseudospectral method, the Gauss pseudospectral method, the Ross-Fahroo
Pseudospectral optimal control
Pseudospectral_optimal_control
Russian mathematician (1930–2010)
quadrature on a sphere to numerical solution of stiff equations, for which he developed explicit Chebyshev methods called DUMKA, systems of PDEs, from domain
Vyacheslav Lebedev (mathematician)
Vyacheslav_Lebedev_(mathematician)
Numerical analysis technique
Michielsen; J. S. Kole; M. T. Figge (2003). "Solving the Maxwell equations by the Chebyshev method: A one-step finite difference time-domain algorithm". IEEE
Finite-difference time-domain method
Finite-difference_time-domain_method
Russian mathematician (1922–2004)
Zbl 0755.47049 P. L. Chebyshev Prize (with Nina Nikolayevna Ural'tseva ) (1966) for the work "Linear and quasilinear equations of elliptic type" USSR
Olga_Ladyzhenskaya
Polynomial sequence
The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials. The Jacobi polynomials
Jacobi_polynomials
Set of points equidistant from a center
sphere in taxicab geometry, and a cube is a sphere in geometry using the Chebyshev distance. The geometry of the sphere was studied by the Greeks. Euclid's
Sphere
Method for estimating new data within known data points
performance in calculation process. As an example we will use points from the equation f ( x ) = sin ( x ) {\displaystyle f(x)=\sin(x)} to demonstrate various
Interpolation
Jacques Charles Chebyshev distance, equation, filter, linkage, polynomials – Pafnuty Chebyshev Chebyshev's inequality (a.k.a. Bienaymé–Chebyshev inequality)
Scientific phenomena named after people
Scientific_phenomena_named_after_people
Mathematical function
equation Γ ( z + 1 ) = z Γ ( z ) . {\displaystyle \Gamma (z+1)=z\Gamma (z).\,} Taking the logarithm on both sides and using the functional equation property
Digamma_function
Concept in dynamical systems
Mitchell Feigenbaum: the solution to the Feigenbaum-Cvitanović functional equation; and the scaling function that described the covers of the attractor of
Feigenbaum_function
Extension of the factorial function
doi:10.1093/IMANUM/12.4.519. Werner, Helmut; Collinge, Robert (1961). "Chebyshev approximations to the Gamma Function". Math. Comput. 15 (74): 195–197
Gamma_function
Study of classical optics using Fourier transforms
different equations than the Helmholtz equation) which give rise to different kinds of orthogonal eigenfunctions such as Legendre polynomials, Chebyshev polynomials
Fourier_optics
transform Hankel transform, the determinant of the Hankel matrix Discrete Chebyshev transform Equivalent, up to a diagonal scaling, to a discrete cosine transform
List_of_transforms
Sequence valued in polynomials
theory as the solutions of certain ordinary differential equations: Laguerre polynomials Chebyshev polynomials Legendre polynomials Zernike polynomials Jacobi
Polynomial_sequence
Measure of the shape of a function
intervals (Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the moments
Moment_(mathematics)
Soviet mathematician
From 1943 he worked at the Mathematical Institute in Moscow, and edited Chebyshev’s complete works. In 1947 he was dismissed from the university and became
Sergei_Bernstein
Characterization of how many integers are prime
}{\frac {\psi (x)}{x}}=1,} where ϑ and ψ are the first and the second Chebyshev functions respectively, and to lim x → ∞ M ( x ) x = 0 , {\displaystyle
Prime_number_theorem
in 1797, predating Jean-Robert Argand by nine years. Arrhenius equation. The equation was first proposed by the Dutch chemist J. H. van 't Hoff in 1884;
List of examples of Stigler's law
List_of_examples_of_Stigler's_law
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