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Jacobi integral | A Wisdom Archive on Jacobi integral | | Jacobi integral A selection of articles related to Jacobi integral | |
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Carl Gustav Jakob Jacobi, Carl Gustav Jakob Jacobi - External link, Jacobian, Jacobi identity, Jacobi's formula, Jacobi symbol, Jacobi integral, Jacobi polynomials
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ARTICLES RELATED TO Jacobi integral | | | |  Jacobi integral: Encyclopedia II - Jacobi's elliptic functions - Definition as inverses of elliptic integralsThe above definition, in terms of the unique meromorphic functions satisfying certain properties, is quite abstract. There is a simpler, but completely equivalent definition, giving the elliptic functions as inverses of the incomplete elliptic integral of the first kind. This is perhaps the easiest definition to understand. Let
Then the elliptic function sn u is given by
See also: Jacobi's elliptic functions, Jacobi's elliptic functions - Introduction, Jacobi's elliptic functions - Notation, Jacobi's elliptic functions - Definition as inverses of elliptic integrals, Jacobi's elliptic functions - Definition in terms of theta functions, Jacobi's elliptic functions - Minor functions, Jacobi's elliptic functions - Addition theorems, Jacobi's elliptic functions - Relations between squares of the functions, Jacobi's elliptic functions - Expansion in terms of the nome Read more here: » Jacobi's elliptic functions: Encyclopedia II - Jacobi's elliptic functions - Definition as inverses of elliptic integrals |
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| | | | Jacobi integral: Encyclopedia II - Jacobi's elliptic functions - Definition in terms of theta functions
Equivalently, Jacobi's elliptic functions can be defined in terms of his theta functions. If we abbreviate as , and respectively as (the theta constants) then the elliptic modulus k is . If we set , we have
Since the Jacobi functions are defined in terms of the elliptic modulus k(τ), we need to invert this and find τ in terms of k. We start from , the complementary modulus. As a fun ...
See also:Jacobi's elliptic functions, Jacobi's elliptic functions - Introduction, Jacobi's elliptic functions - Notation, Jacobi's elliptic functions - Definition as inverses of elliptic integrals, Jacobi's elliptic functions - Definition in terms of theta functions, Jacobi's elliptic functions - Minor functions, Jacobi's elliptic functions - Addition theorems, Jacobi's elliptic functions - Relations between squares of the functions, Jacobi's elliptic functions - Expansion in terms of the nome Read more here: » Jacobi's elliptic functions: Encyclopedia II - Jacobi's elliptic functions - Definition in terms of theta functions |
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| | | | Jacobi integral: Encyclopedia II - Jacobi's elliptic functions - Addition theorems The functions satisfy the two algebraic relations
From this we see that (cn, sn, dn) parametrizes an elliptic curve which is the intersection of the two quadrics defined by the above two equations. We now may define a group law for points on this curve by the addition formulas for the Jacobi functions
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See also:Jacobi's elliptic functions, Jacobi's elliptic functions - Introduction, Jacobi's elliptic functions - Notation, Jacobi's elliptic functions - Definition as inverses of elliptic integrals, Jacobi's elliptic functions - Definition in terms of theta functions, Jacobi's elliptic functions - Minor functions, Jacobi's elliptic functions - Addition theorems, Jacobi's elliptic functions - Relations between squares of the functions, Jacobi's elliptic functions - Expansion in terms of the nome Read more here: » Jacobi's elliptic functions: Encyclopedia II - Jacobi's elliptic functions - Addition theorems |
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| | | | Jacobi integral: Encyclopedia II - Jacobi's elliptic functions - Minor functions It is conventional to denote the reciprocals of the three functions above by reversing the order of the two letters of the function name:
The ratios of the three primary functions are denoted by the first letter of the numerator followed by the first letter of the denominator:
More compactly, we can write
where p, q, and r are any of the ...
See also:Jacobi's elliptic functions, Jacobi's elliptic functions - Introduction, Jacobi's elliptic functions - Notation, Jacobi's elliptic functions - Definition as inverses of elliptic integrals, Jacobi's elliptic functions - Definition in terms of theta functions, Jacobi's elliptic functions - Minor functions, Jacobi's elliptic functions - Addition theorems, Jacobi's elliptic functions - Relations between squares of the functions, Jacobi's elliptic functions - Expansion in terms of the nome Read more here: » Jacobi's elliptic functions: Encyclopedia II - Jacobi's elliptic functions - Minor functions |
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| | | | Jacobi integral: Encyclopedia II - Jacobi's elliptic functions - Relations between squares of the functions where m + m1 = 1 and m = k2.
Additional relations between squares can be obtained by noting that and that where p,q,r are any of the letters s,c,d,n and ss=cc=dd=nn=1.
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See also:Jacobi's elliptic functions, Jacobi's elliptic functions - Introduction, Jacobi's elliptic functions - Notation, Jacobi's elliptic functions - Definition as inverses of elliptic integrals, Jacobi's elliptic functions - Definition in terms of theta functions, Jacobi's elliptic functions - Minor functions, Jacobi's elliptic functions - Addition theorems, Jacobi's elliptic functions - Relations between squares of the functions, Jacobi's elliptic functions - Expansion in terms of the nome Read more here: » Jacobi's elliptic functions: Encyclopedia II - Jacobi's elliptic functions - Relations between squares of the functions |
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| | | | Jacobi integral: Encyclopedia II - Jacobi's elliptic functions - Introduction There are twelve Jacobian elliptic functions. Each of the twelve corresponds to an arrow drawn from one corner of a rectangle to another. Each of the corners of the rectangle are labeled, by convention, s, c, d and n. The rectangle is understood to be lying on the complex plane, so that s is at the origin, c is at the point K on the real axis, d is at the point K +iK' and n is at point iK' on the imaginary axis. The numbers K and K' are called the quarter periods. The twelve Jacobian elliptic function ...
See also:Jacobi's elliptic functions, Jacobi's elliptic functions - Introduction, Jacobi's elliptic functions - Notation, Jacobi's elliptic functions - Definition as inverses of elliptic integrals, Jacobi's elliptic functions - Definition in terms of theta functions, Jacobi's elliptic functions - Minor functions, Jacobi's elliptic functions - Addition theorems, Jacobi's elliptic functions - Relations between squares of the functions, Jacobi's elliptic functions - Expansion in terms of the nome Read more here: » Jacobi's elliptic functions: Encyclopedia II - Jacobi's elliptic functions - Introduction |
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| | | | Jacobi integral: Encyclopedia II - Theta function - Product representations The Jacobi theta function can be expressed as a product, through the Jacobi triple product theorem:
The auxiliary functions have the expressions, with q = expiπτ:
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See also:Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function Read more here: » Theta function: Encyclopedia II - Theta function - Product representations |
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| | | | Jacobi integral: Encyclopedia II - Theta function - Jacobi identities Jacobi's identities describe how theta functions transform under the modular group. Let
Then
See also: proof of Jacobi's identity for functions on PlanetMath.. Note that the conventions for z in that reference differ from those here by a factor of π.
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See also:Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function Read more here: » Theta function: Encyclopedia II - Theta function - Jacobi identities |
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| | | | Jacobi integral: Encyclopedia II - Theta function - Relation to the Weierstrass elliptic function The theta function was used by Jacobi to construct (in a form adapted to easy calculation) his elliptic functions as the quotients of the above four theta functions, and could have been used by him to construct Weierstrass's elliptic functions also, since
where the second derivative is with respect to z and the constant c is defined so that the Laurent expansion of at z = 0 has zero constant term.
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See also:Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function Read more here: » Theta function: Encyclopedia II - Theta function - Relation to the Weierstrass elliptic function |
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| | | | Jacobi integral: Encyclopedia II - Theta function - A solution to heat equation The Jacobi theta function is the unique solution to the one-dimensional heat equation with periodic boundary conditions at time zero. This is most easily seen by taking z = x to be real, and taking τ = it with t real and positive. Then we can write
which solves the heat equation
That this solution is unique can be seen by noting that at t = 0, the theta function becomes the Dirac comb:
where δ is the Dirac delta function. Thus, general solution can be specified by convolving the (periodi ...
See also:Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function Read more here: » Theta function: Encyclopedia II - Theta function - A solution to heat equation |
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| | | | Jacobi integral: Encyclopedia II - Theta function - Jacobi theta function The Jacobi theta function is a function defined for two complex variables z and τ, where z can be any complex number and τ is confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula
If τ is fixed, this becomes a Fourier series for a periodic entire function of z with period 1; in this case, the theta function satisfies the identity
The function also behaves very regularly with respect to addition by τ and satisfies the functional equation
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See also:Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function Read more here: » Theta function: Encyclopedia II - Theta function - Jacobi theta function |
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| | | | Jacobi integral: Encyclopedia II - Theta function - Auxiliary functions It is convenient to define three auxiliary theta functions, which we may write
This notation follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome q = exp(πτ) rather than τ, and theta there is called θ3, with termed θ0, named θ2, and c ...
See also:Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function Read more here: » Theta function: Encyclopedia II - Theta function - Auxiliary functions |
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| | | | Jacobi integral: Encyclopedia II - Theta function - Relation to the Riemann zeta function The relation
was used by Riemann to prove the functional equation for Riemann's zeta function, by means of the integral
which can be shown to be invariant under substitution of s by 1 − s. The corresponding integral for z not zero is given in the article on the Hurwitz zeta function.
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See also:Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function Read more here: » Theta function: Encyclopedia II - Theta function - Relation to the Riemann zeta function |
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| | | | Jacobi integral: Encyclopedia II - Elliptic integral - Incomplete elliptic integral of the first kind The incomplete elliptic integral of the first kind F is defined, in Jacobi's form, as
Equivalently, using alternate notation,
where it is understood that when there is a vertical bar used, the argument following the vertical bar is the parameter (as defined above), and, when a backslash is used, it is followed by the modular angle. Note that
F(x;k) = u
with u as defined ...
See also:Elliptic integral, Elliptic integral - Notation, Elliptic integral - Incomplete elliptic integral of the first kind, Elliptic integral - Incomplete elliptic integral of the second kind, Elliptic integral - Incomplete elliptic integral of the third kind, Elliptic integral - Complete elliptic integral of the first kind, Elliptic integral - Complete elliptic integral of the second kind, Elliptic integral - History Read more here: » Elliptic integral: Encyclopedia II - Elliptic integral - Incomplete elliptic integral of the first kind |
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| | | | Jacobi integral: Encyclopedia II - Theta function - Relation to the Heisenberg group The Jacobi theta function can be thought of as belonging to a representation of the Heisenberg group in quantum mechanics, sometimes called the theta representation. This can be seen by explicitly constructing the group. Let f(z) be a holomorphic function, let a and b be real numbers, and fix a value of τ. Then define the operators Sa and Tb such that
(SaSee also: Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function Read more here: » Theta function: Encyclopedia II - Theta function - Relation to the Heisenberg group |
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| | | | Jacobi integral: Encyclopedia II - Abelian variety - History and motivation The success in the early nineteenth century of the theory of elliptic functions in giving a basis for the theory of elliptic integrals left open an obvious avenue of research. The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials. When those were replaced by polynomials of higher degree, say quintics, what would happen?
In the work of Niels Abel and Carl Jacobi, the answer was formulated: this would involve functions of two complex variables, having four independent periods (i.e. p ...
See also:Abelian variety, Abelian variety - History and motivation, Abelian variety - Analytic theory, Abelian variety - Definition, Abelian variety - Riemann conditions, Abelian variety - The Jacobian of an algebraic curve, Abelian variety - Abelian functions, Abelian variety - Algebraic definition, Abelian variety - Structure of the group of points, Abelian variety - Polarization and dual abelian variety, Abelian variety - Dual abelian variety, Abelian variety - Polarizations, Abelian variety - Polarizations over the complex numbers, Abelian variety - Abelian scheme Read more here: » Abelian variety: Encyclopedia II - Abelian variety - History and motivation |
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| | | | Jacobi integral: Encyclopedia II - Orthogonal polynomials - Introduction In mathematics, an orthogonal polynomial sequence is an infinite sequence of polynomials p0(x), p1(x), p2(x) ... , in which each pn(x) has degree n, and such that any two different polynomials in the sequence are orthogonal to each other in the following sense:
One can define an inner product on functions, (analogous to the ordinary "dot product" for vectors), by integrating the product of ...
See also:Orthogonal polynomials, Orthogonal polynomials - Introduction, Orthogonal polynomials - General properties of orthogonal polynomial sequences, Orthogonal polynomials - Recurrence relations, Orthogonal polynomials - Existence of real roots, Orthogonal polynomials - Interlacing of roots, Orthogonal polynomials - Differential equations leading to orthogonal polynomials, Orthogonal polynomials - Rodrigues formula, Orthogonal polynomials - The numbers λn, Orthogonal polynomials - Second form for the differential equation, Orthogonal polynomials - Third form for the differential equation, Orthogonal polynomials - Formulas involving derivatives, Orthogonal polynomials - The classical orthogonal polynomials, Orthogonal polynomials - Jacobi polynomials, Orthogonal polynomials - Gegenbauer polynomials, Orthogonal polynomials - Legendre polynomials, Orthogonal polynomials - Chebyshev polynomials, Orthogonal polynomials - Laguerre polynomials, Orthogonal polynomials - Hermite polynomials, Orthogonal polynomials - Table of classical orthogonal polynomials Read more here: » Orthogonal polynomials: Encyclopedia II - Orthogonal polynomials - Introduction |
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