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In mathematics, a quadratic integral is an integral of the form ∫ d x a + b x + c x 2 . {\displaystyle \int {\frac {dx}{a+bx+cx^{2}}}.} It can be evaluated
Quadratic_integral
Polynomial with all terms of degree two
particular, in the theory of quadratic fields, continued fractions, and modular forms. The theory of integral quadratic forms in n variables has important
Quadratic_form
Topics referred to by the same term
that is a root of a quadratic polynomial Quadratic integral, the integral of the reciprocal of a second-degree polynomial Quadratic form (statistics),
Quadratic
Root of a quadratic polynomial with a unit leading coefficient
Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations, and other questions related to integral quadratic forms
Quadratic_integer
Formula that provides the solutions to a quadratic equation
algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations,
Quadratic_formula
Quadratic homogeneous polynomial in two variables
in quadratic form. A quadratic form with integer coefficients is called an integral binary quadratic form, often abbreviated to binary quadratic form
Binary_quadratic_form
Calculus on stochastic processes
Stratonovich integral can readily be expressed in terms of the Itô integral, and vice versa. The main benefit of the Stratonovich integral is that it obeys
Stochastic_calculus
Calculus of stochastic differential equations
formulas of standard calculus, due to quadratic variation terms. This can be contrasted to the Stratonovich integral as an alternative formulation; it does
Itô_calculus
Operation in calculus
better approximations to the integral, can be carried further: Simpson's rule approximates the integrand by a piecewise quadratic function. Riemann sums, the
Integral
the integral sign Trigonometric substitution Partial fractions in integration Quadratic integral Proof that 22/7 exceeds π Trapezium rule Integral of the
List_of_calculus_topics
Quantity defined for a stochastic process
mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is
Quadratic_variation
Unsolved problem in geometry
imaginary quadratic field. In the latter case, the Hodge conjecture is only known in special cases. Hodge's original conjecture was: Integral Hodge conjecture
Hodge_conjecture
Gives conditions for the solvability of quadratic equations modulo prime numbers
theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime
Quadratic_reciprocity
On when an integer positive definite quadratic form represents all positive integers
published in 2000. Conway conjectured an analogous statement for integral quadratic forms, with the constant 15 replaced by 290. Bhargava and Jonathan
15_and_290_theorems
Special function defined by an integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied
Elliptic_integral
Mathematical element
This example and the previous one are examples of quadratic integers. The integral closure of a quadratic extension Q ( d ) {\displaystyle \mathbb {Q} ({\sqrt
Integral_element
Method for solving quadratic equations
elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form a x 2 + b x + c {\displaystyle \textstyle ax^{2}+bx+c}
Completing_the_square
Function of the coefficients of a polynomial that gives information on its roots
useful in the study of quadratic fields is the fundamental discriminant. It arises in the theory of integral binary quadratic forms, which are expressions
Discriminant
Mathematical concept
genus is a classification of quadratic forms and lattices over the ring of integers. An integral quadratic form is a quadratic form on Zn, or equivalently
Genus_of_a_quadratic_form
Formulation of quantum mechanics
The path-integral formulation of quantum mechanics generalizes the action principle of classical mechanics. It replaces the classical notion of a single
Path-integral_formulation
Unique algebraic expression given by Srinivasa Ramanujan
branch of mathematics, Ramanujan's ternary quadratic form is the algebraic expression x2 + y2 + 10z2 with integral values for x, y and z. Srinivasa Ramanujan
Ramanujan's ternary quadratic form
Ramanujan's_ternary_quadratic_form
Special state of wave and quantum systems in physics
^{2}r{\text{d}}r} diverges. Let's try to change the wave function so that the quadratic integral converges and the potential varies near -1. Consider the following
Bound_state_in_the_continuum
by Carl Friedrich Gauss, for performing a binary operation on integral binary quadratic forms (IBQFs). Gauss presented this rule in his Disquisitiones
Gauss_composition_law
American-Turkish mathematician (born 1992)
2025, Alpöge, Bhargava, Wei Ho, and Ari Shnidman proved that for any quadratic extension of number fields K / F {\displaystyle K/F} there exists an abelian
Levent_Alpöge
Square root of the mean square
S x {\displaystyle \mathrm {RMS} _{x}} . The RMS is also known as the quadratic mean (denoted M 2 {\displaystyle M_{2}} ), a special case of the generalized
Root_mean_square
Approximation of a function by a polynomial
function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving
Taylor's_theorem
Mathematical concept
mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to
Ε-quadratic_form
Conjecture on zeros of the zeta function
that the generalized Riemann hypothesis implies that Ramanujan's integral quadratic form x2 + y2 + 10z2 represents all integers that it represents locally
Riemann_hypothesis
Hypercomplex number system
0), (5 6 1), (6 0 2), (0 1 3). These are the nonzero codewords of the quadratic residue code of length 7 over the Galois field of two elements, GF(2)
Octonion
Russian mathematician
general linear group and in 1983 in Warsaw with talk Integral representation of quadratic forms by quadratic forms: multiplicative properties. He held visiting
Anatoli_N._Andrianov
Mathematical function
functions arise by composing the exponential function with a concave quadratic function: f ( x ) = exp ( α x 2 + β x + γ ) , {\displaystyle f(x)=\exp
Gaussian_function
Lattice in 8-dimensional space with special properties
by 2) is a positive definite even unimodular quadratic form in 8 variables, and conversely such a quadratic form can be used to construct a positive-definite
E8_lattice
reduced to a system of equations with constant coefficients and have a quadratic integral of definite sign”. The Chetaev's theorem generalizes the Lagrange's
Nikolay_Gur'yevich_Chetaev
Lebesgue-Stieltjes integration
arises from the quadratic covariation of U and V. (The earlier result can then be seen as a result pertaining to the Stratonovich integral.) When g(x) =
Lebesgue–Stieltjes integration
Lebesgue–Stieltjes_integration
Mathematical constant
In mathematical analysis and number theory, Somos' quadratic recurrence constant or simply Somos' constant is a constant defined as an expression of infinitely
Somos' quadratic recurrence constant
Somos'_quadratic_recurrence_constant
Mathematical relation assigning a probability event to a cost
regression theory, which is based on the quadratic loss function. The quadratic loss function is also used in linear-quadratic optimal control problems. In these
Loss_function
Commutative ring with no zero divisors other than zero
mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. In an integral domain, every
Integral_domain
Algebra based on a vector space with a quadratic form
a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure
Clifford_algebra
Pictorial representation of the behavior of subatomic particles
matrix in the quadratic part of the action in both the Bose and Fermi case. For real Grassmann fields, for Majorana fermions, the path integral is a Pfaffian
Feynman_diagram
Function defined by a hypergeometric series
then there is a quadratic transformation of the hypergeometric function, connecting it to a different value of z related by a quadratic equation. The first
Hypergeometric_function
In mathematical physics and harmonic analysis, the quadratic Fourier transform is an integral transform that generalizes the fractional Fourier transform
Quadratic_Fourier_transform
Curve used in computer graphics and related fields
Pn, where n is called the order of the curve (n = 1 for linear, 2 for quadratic, 3 for cubic, etc.). The first and last control points are always the
Bézier_curve
Mathematical operation
the second derivative is related to the best quadratic approximation for a function f. This is the quadratic function whose first and second derivatives
Second_derivative
Mathematical transform that expresses a function of time as a function of frequency
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent
Fourier_transform
Topics referred to by the same term
indefinite pronoun Indefinite integral, another name for the antiderivative Indefinite forms in algebra: see definite quadratic forms an indefinite matrix
Indefinite
Relating coefficients and roots of a polynomial
Vieta's formulas applied to quadratic and cubic polynomials: The roots r 1 , r 2 {\displaystyle r_{1},r_{2}} of the quadratic polynomial P ( x ) = a x 2
Vieta's_formulas
Methods of calculating definite integrals
family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more
Numerical_integration
In number theory, measure of non-unique factorization
formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as put into something like a final form
Ideal_class_group
Classify quadratic forms over algebraic number fields
problem: to solve a given quadratic equation with algebraic numerical coefficients in any number of variables by integral or fractional numbers belonging
Hilbert's_eleventh_problem
Young integral and Young differential equations and makes heavy use of the concept of p-variation. p-variation should be contrasted with the quadratic variation
P-variation
Type of stochastic process
a consequence of the integration by parts formula for the Itō integral. The quadratic variation exists for every semimartingale. The class of semimartingales
Semimartingale
Complex number whose real and imaginary parts are both integers
Gaussian integers is the integral closure of the integers in the Gaussian rationals. This implies that Gaussian integers are quadratic integers and that a
Gaussian_integer
Mathematical proof technique
Vieta jumping is a classical method in the theory of quadratic Diophantine equations and binary quadratic forms. For example, it was used in the analysis of
Vieta_jumping
Product of a number by itself
be used in place of x2. The adjective which corresponds to squaring is quadratic. The square of an integer may also be called a square number or a perfect
Square_(algebra)
Method for numerical integration
interpolating quadratic polynomial plus an arbitrarily scaled cubic polynomial that vanishes at all three points in the interval, and the integral of this second
Simpson's_rule
Term in an algebraic expression which does not contain any variables
contain any variables and therefore is constant. For example, in the quadratic polynomial x 2 + 2 x + 3 {\displaystyle x^{2}+2x+3} , the number 3 is
Constant_term
Invariant of a quadratic form over a field of characteristic 2
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician Cahit Arf (1941)
Arf_invariant
(quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism groups. The mass formula is often given for integral quadratic
Smith–Minkowski–Siegel mass formula
Smith–Minkowski–Siegel_mass_formula
Procedure to solve equations of second degree
In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is a x 2 + b x + c = 0 , {\displaystyle ax^{2}+bx+c=0
Solving quadratic equations with continued fractions
Solving_quadratic_equations_with_continued_fractions
American mathematician
doi:10.2307/2033023. JSTOR 2033023. O'Meara, O. T. (1957). "Integral equivalence of quadratic forms in ramified local fields". American Journal of Mathematics
O._Timothy_O'Meara
Number whose square is a given number
major use in the formula for solutions of a quadratic equation. Quadratic fields and rings of quadratic integers, which are based on square roots, are
Square_root
Quadratic imaginary number fields with unique factorisation
theorem or Stark-Heegner theorem establishes the complete list of the quadratic imaginary number fields whose rings of integers are principal ideal domains
Stark–Heegner_theorem
Nearest integers from a number
Gauss introduced the square bracket notation [x] in his third proof of quadratic reciprocity (1808). This remained the standard in mathematics until Kenneth
Floor_and_ceiling_functions
large number of closed-form approximations and expressions in terms of integrals or series have been given for the perimeter of an ellipse. An ellipse
Perimeter_of_an_ellipse
Class of integrals appearing in quantum field theory
In quantum field theory and statistical mechanics, loop integrals are the integrals which appear when evaluating the Feynman diagrams with one or more
Loop_integral
1929 mathematical conjecture
semisimple Lie groups. Meyer's theorem states that an indefinite integral quadratic form Q in n variables, n ≥ 5, nontrivially represents zero, i.e. there
Oppenheim_conjecture
Function whose squared absolute value has finite integral
In mathematics, a square-integrable function, also called a quadratically integrable function or L 2 {\displaystyle L^{2}} function or square-summable
Square-integrable_function
theorem of calculus gives the relationship between differential calculus and integral calculus. The names are mostly traditional, so that for example the fundamental
List of theorems called fundamental
List_of_theorems_called_fundamental
In algebra, element without non-trivial factors
In algebra, an irreducible element of an integral domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product
Irreducible_element
Type of algebras, possibly non associative
necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies N ( x y ) = N ( x ) N ( y ) {\displaystyle N(xy)=N(x)N(y)}
Composition_algebra
Branch of differential geometry
geometry, the quadratic form is positive definite. Relaxing this condition, and allowing that some non-zero vectors can be null under the quadratic form allows
Riemannian_geometry
Integration technique using recurrence relations
In integral calculus, integration by reduction formulae is a method relying on recurrence relations. It is used when an expression containing an integer
Integration by reduction formulae
Integration_by_reduction_formulae
Stochastic process generalizing Brownian motion
(local) martingale W with W0 = 0 is a Wiener process if and only if its quadratic variation is [W, W]t = t (which means that Wt2 − t is a (local) martingale)
Wiener_process
How many times curves wind around each other
The linking number was introduced by Gauss in the form of the linking integral. It is an important object of study in knot theory, algebraic topology
Linking_number
Method for estimating new data within known data points
Online tools for linear Archived 2016-09-18 at the Wayback Machine, quadratic Archived 2016-09-18 at the Wayback Machine, cubic spline Archived 2016-08-20
Interpolation
Algebra with unique prime factorization
Leedham-Green showed that such an R may be constructed as the integral closure of a PID in a quadratic field extension. In 1976, M. Rosen showed how to realize
Dedekind_domain
Condition under which an odd prime is a sum of two squares
Lagrange completed a proof in 1775 based on his general theory of integral quadratic forms. The following presentation incorporates a slight simplification
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Relationship between the rational roots of a polynomial and its extreme coefficients
solution r, then factoring out (x – r) leaves a quadratic polynomial whose two roots, found with the quadratic formula, are the remaining two roots of the
Rational_root_theorem
American mathematician
Schulze-Pillot, R. (1993) Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Duke Mathematical
William_Duke_(mathematician)
Positive real number which when multiplied by itself gives 5
{\displaystyle -{\sqrt {5}}} , it solves the quadratic equation x 2 − 5 = 0 {\displaystyle x^{2}-5=0} , making it a quadratic integer, a type of algebraic number
Square_root_of_5
of the minimal polynomial of α. Examples of monogenic fields include: Quadratic fields: if K = Q ( d ) {\displaystyle K=\mathbf {Q} ({\sqrt {d}})} with
Monogenic_field
theorem Primitive root modulo n Multiplicative order Discrete logarithm Quadratic residue Euler's criterion Legendre symbol Gauss's lemma (number theory)
List_of_number_theory_topics
Field of mathematics and science based on non-linear systems and initial conditions
showed that, at least for dissipative and conservative quadratic systems, three-dimensional quadratic systems with only three or four terms on the right-hand
Chaos_theory
Polynomial equation, generally univariate
Babylonian mathematicians, as early as 2000 BC could solve some kinds of quadratic equations (displayed on Old Babylonian clay tablets). Univariate algebraic
Algebraic_equation
Complex number that solves a monic polynomial with integer coefficients
algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some
Algebraic_integer
Natural number
often normalized by the condition that they have integral one, maximum value one, or square integral one, depending on the application. 1 is the most
1
Algebraic construction
is the corresponding quadratic field, then O K {\displaystyle {\mathcal {O}}_{K}} is a ring of quadratic integers and its integral basis is given by (
Ring_of_integers
product integral . product rule . proper fraction . proper rational function . Pythagorean theorem . Pythagorean trigonometric identity . quadratic function
Glossary_of_calculus
Type of integral domain
the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero
Unique_factorization_domain
Measure of variation in statistics
_{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}},} The error in this approximation decays quadratically (as 1/N2), and it is suited for all but the smallest samples or highest
Standard_deviation
(t_{*})\cap L_{0}\neq 0} , the sign is determined by the crossing form, a quadratic form on the intersection γ ( t ∗ ) ∩ L 0 {\displaystyle \gamma (t_{*})\cap
Maslov_index
Stack Exchange mathematician
solution. His approach involved reducing an eighth-degree polynomial to a quadratic equation through symmetry analysis and deriving the golden ratio from
Cleo_(mathematician)
Indefinite quadratic forms in > 4 variables over the rationals nontrivially represent 0
In number theory, Meyer's theorem on quadratic forms states that an indefinite quadratic form Q in five or more variables over the field of rational numbers
Meyer's_theorem
Algorithm for finding zeros of functions
Furthermore, for a root of multiplicity 1, the convergence is at least quadratic (see Rate of convergence) in some sufficiently small neighbourhood of
Newton's_method
German polymath and scholar (1777–1855)
made numerous contributions, such as the composition law, the law of quadratic reciprocity, and proved the triangular case of the Fermat polygonal number
Carl_Friedrich_Gauss
distribution theory of holomorphic functions Line integral Cauchy's integral theorem Cauchy's integral formula Residue theorem Liouville's theorem (complex
List of complex analysis topics
List_of_complex_analysis_topics
Theorem in mathematics
Reynolds Integral Lists of integrals Integral transform Leibniz integral rule Definitions Antiderivative Integral (improper) Riemann integral Lebesgue
Inverse_function_theorem
Measure for evaluating probabilistic forecasts
_{b}(p_{i})} is strictly proper for all b > 1 {\displaystyle b>1} . The quadratic scoring rule is a strictly proper scoring rule S Q ( p , i ) = 2 p i −
Scoring_rule
Quickly converging computation of π
69399375105820974944592307816406286208998625\dots } The algorithm has quadratic convergence, which essentially means that the number of correct digits
Gauss–Legendre_algorithm
Mathematical method in calculus
by parts for the Lebesgue–Stieltjes integral Integration by parts for semimartingales, involving their quadratic covariation. Integration by substitution
Integration_by_parts
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