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Encyclopedia > Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. This article deals with the concept of an integral in calculus. ... For other uses, see Curve (disambiguation). ... The ellipse and some of its mathematical properties. ... Giulio Carlo, Count Fagnano, and Marquis de Toschi, (December 6, 1682 Sinigaglia - September 26, 1766) was an Italian mathematician. ... Euler redirects here. ...


In the modern definition, an elliptic integral is any function f which can be expressed in the form Partial plot of a function f. ...

f(x) = int_{c}^{x} R(t,P(t)) dt ,!

where R is a rational function of its two arguments, P is the square root of a polynomial of degree 3 or 4 (a cubic or quartic) with no repeated roots, and c is a constant. In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...


In general, elliptic integrals cannot be expressed in terms of elementary functions; exceptions to this are when P does have repeated roots, or when R(x,y) contains no odd powers of y. However, with appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions, and the three canonical forms (i.e. the elliptic integrals of the first, second and third kind).


Besides the forms given below, the elliptic integrals may also be expressed in Legendre form and Carlson symmetric form. Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz-Christoffel mapping. In mathematics, the Legendre forms of elliptic integrals, F(φ,k), E(φ,k) and P(φ,k,n) are defined by and Categories: Math stubs ... In mathematics, the Carlson symmetric forms of elliptic integrals, RC(x,y), RD(x,y,z), RF(x,y,z) and RJ(x,y,z,p) are defined by Categories: Special functions | Elliptic functions | Math stubs ... In complex analysis, a discipline within mathematics, a Schwarz-Christoffel mapping is a transformation of the complex plane that maps the upper half-plane conformally to a polygon. ...

Contents

Notation

Elliptic integrals are often expressed as functions of a variety of different arguments. These different arguments are completely equivalent (they give the same elliptic integral), but can be confusing due to their different appearance. Most texts adhere to a canonical naming scheme. Before defining the integrals, we review the naming conventions for the arguments:

Note that the above three are completely determined by one another; specifying one is the same as specifying another. The elliptic integrals will also depend on another argument; this can also be specified in a number of different ways: In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and eccentricity. ...

Specifying any one of these determines the others, and thus again, these may be used interchangeably in the notation. Note that u also depends on m. Some additional relationships involving u include In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ...

cos phi = textrm{cn}; u,!

and

sqrt{1-msin^2 phi} = textrm{dn}; u.,!

The latter is sometimes called the delta amplitude and written as Delta(phi)=textrm{dn}; u,!.


Sometimes the literature refers to the complementary parameter, the complementary modulus or the complementary modular angle. These are further defined in the article on quarter periods. In mathematics, the quarter periods K(m) and iK′(m) are special functions that appear in the theory of elliptic functions. ...


Incomplete elliptic integral of the first kind

The incomplete elliptic integral of the first kind F is defined as

F(phisetminus o!varepsilon ) = F(phi|m) = int_0^phifrac{dtheta}{sqrt{1-(sinthetasin o!varepsilon)^2}},!

Equivalently, using notation in Jacobi's form, one sets x=sin phi ~,~ t=sin theta;!; then Karl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (December 10, 1804 - February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p. ...

F(phisetminus o!varepsilon ) = F(x;k) = int_{0}^{x} frac{dt}{sqrt{(1-t^2)(1-k^2 t^2)} },!

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. In this sense, F(sinphi;sin o!varepsilon) = F(phi|sin (o!varepsilon)^2) = F(phisetminus o!varepsilon )~ ,!. These notations are borrowed from the book Abramowitz and Stegun; the use of the delimiters ; | is traditional in elliptic integrals. Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ...


There are differing conventions regarding notation of elliptic integrals. The differences can be very confusing, especially to a novice, see [[1]]. The functions, which evaluate the elliptic integrals, do not have standard and unic names and meanings (like sqrt, sin and erf have). Different notations are used in the literature. Gradstein, Ryzhik [[2], Eq.(8.111)] and the wiki article Legendre form use notation F(phi,k) ,!, which is equivalent to our F(phi|k^2)~ ,!; also E(phi,k)=E(phi|k^2)~ ,! below. For example, if one translates code from the language of Mathematica into the language of Maple, one should replace the argument of the EllipticK function by its square root, and correspondingly, in the translation from Maple to Mathematica the argument should be replaced by its square. EllipticK(x) in Maple is almost equivalent of EllipticK[x^2] in Mathematica; one may expect to get the same result in both systems, at least while 0<x<1. In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. ... Sin is a term used mainly in a religious context to describe an act that violates a moral code of conduct or the state of having committed such a violation. ... In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ... In mathematics, the Legendre forms of elliptic integrals, F(&#966;,k), E(&#966;,k) and P(&#966;,k,n) are defined by and Categories: Math stubs ... This article is about computer software. ... Maple 9. ... In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. ... In algebra, the square of x is written x2 and is defined as the product of x with itself: x × x. ...


Note that

F(x;k) = u ,!

with u as defined above: thus, the Jacobian elliptic functions are inverses to the elliptic integrals. In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ...


Incomplete elliptic integral of the second kind

The incomplete elliptic integral of the second kind E is

E(phisetminus o!varepsilon) = E(phi|m) = int_0^phisqrt{1-(sinthetasin o!varepsilon)^2} dtheta.,!

Equivalently, using alternate notation (substituting t=sintheta,!),

E(x;k) = int_{0}^{x} frac{sqrt{1-k^2 t^2} }{sqrt{1-t^2}} dt. ,!

Additional relations include

E(phi|m) = int_0^u textrm{dn}^2 w ;dw = u-mint_0^u textrm{sn}^2 w ;dw = (1-m)u+mint_0^u textrm{cn}^2 w ;dw.,!

Incomplete elliptic integral of the third kind

The incomplete elliptic integral of the third kind Pi,! is

Pi(n; phi|m) = int_0^phi frac{1}{1-nsin^2 theta} frac {dtheta}{sqrt{1-(sinthetasin o!varepsilon)^2}},,!

or

Pi(n; phi|m) = int_{0}^{x} frac{1}{1-nt^2} frac{dt}{sqrt{(1-k^2 t^2)(1-t^2) }},,!

or

Pi(n; phi|m) = int_0^u frac{dw}{1-n textrm{sn}^2 (w|m)}. ; ,!

The number n is called the characteristic and can take on any value, independently of the other arguments. Note though that the value Pi(1;pi/2|m),! is infinite, for any m,!.


Complete elliptic integral of the first kind

Main article: complete elliptic integral of the first kind. The complete elliptic integral of the first kind K may be defined as or and can be computed in terms of the arithmetic-geometric mean. ...


Complete elliptic integral of the second kind

Main article: complete elliptic integral of the second kind. The complete elliptic integral of the second kind E may be defined as or It is a special case of the incomplete elliptic integral of the second kind: Category: ...


History

Historically, elliptic functions were discovered as inverse functions of elliptic integrals, and this one in particular: we have F(sn(z;k);k) = z where sn is one of Jacobi's elliptic functions. In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ...


See also

In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In mathematics, the Legendre forms of elliptic integrals, F(&#966;,k), E(&#966;,k) and P(&#966;,k,n) are defined by and Categories: Math stubs ... In mathematics, the Carlson symmetric forms of elliptic integrals, RC(x,y), RD(x,y,z), RF(x,y,z) and RJ(x,y,z,p) are defined by Categories: Special functions | Elliptic functions | Math stubs ... In complex analysis, a discipline within mathematics, a Schwarz-Christoffel mapping is a transformation of the complex plane that maps the upper half-plane conformally to a polygon. ...

References


  Results from FactBites:
 
NationMaster - Encyclopedia: Elliptic integral (0 words)
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler.
The complete elliptic integral of the second kind E may be defined as or It is a special case of the incomplete elliptic integral of the second kind: Category:...
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Fagnano and Leonhard Euler.
elliptic integral standard forms@Everything2.com (0 words)
The actual method for transforming a general elliptic integral into these standard forms is actually quite involved, and requires a lot of fairly tedious algebra, but before automatic computers were widely available for performing numerical quadrature this was the best way to proceed.
Once the integral has been thus reduced into elementary integrals and the Legendre standard forms, algorithms involving Landen's transformation and the arithmetic geometric mean are available to calculate a definite integral for the entire expression.
Tables giving the values of the integral for various values of the amplitude (the upper limit of the integrand), the modulus, and in the case of integrals of the third kind, the parameter are also available.
  More results at FactBites »

 
 

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