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 by Giulio Fagnano and Leonhard Euler (c. 1750). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse.
Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form
where R is a rational function of its two arguments, P is a polynomial of degree 3 or 4 with no repeated roots, and c is a constant.
In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when P has repeated roots, or when R(x, y) contains no odd powers of y or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms (i.e. the elliptic integrals of the first, second and third kind).
Besides the Legendre form given below, the elliptic integrals may also be expressed in Carlson symmetric form. Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz–Christoffel mapping. Historically, elliptic functions were discovered as inverse functions of elliptic integrals.
Argument notation
Incomplete elliptic integrals are functions of two arguments; complete elliptic integrals are functions of a single argument. These arguments are expressed in a variety of different but equivalent ways (they give the same elliptic integral). Most texts adhere to a canonical naming scheme, using the following naming conventions.
For expressing one argument:
α, the modular angle
k = sin α, the elliptic modulus or eccentricity
m = k2 = sin2α, the parameter
Each of the above three quantities is completely determined by any of the others (given that they are non-negative). Thus, they can be used interchangeably.
The other argument can likewise be expressed as φ, the amplitude, or as x or u, where x = sin φ = sn u and sn is one of the Jacobian elliptic functions.
Specifying the value of any one of these quantities determines the others. Note that u also depends on m. Some additional relationships involving u include
The latter is sometimes called the delta amplitude and written as Δ(φ) = dn u. Sometimes the literature also refers to the complementary parameter, the complementary modulus, or the complementary modular angle. These are further defined in the article on quarter periods.
In this notation, the use of a vertical bar as delimiter indicates that the argument following it is the "parameter" (as defined above), while the backslash indicates that it is the modular angle. The use of a semicolon implies that the argument preceding it is the sine of the amplitude: This potentially confusing use of different argument delimiters is traditional in elliptic integrals and much of the notation is compatible with that used in the reference book by Abramowitz and Stegun and that used in the integral tables by Gradshteyn and Ryzhik.
There are still other conventions for the notation of elliptic integrals employed in the literature. The notation with interchanged arguments, F(k, φ), is often encountered; and similarly E(k, φ) for the integral of the second kind. Abramowitz and Stegun substitute the integral of the first kind, F(φ, k), for the argument φ in their definition of the integrals of the second and third kinds, unless this argument is followed by a vertical bar: i.e. E(F(φ, k) | k2) for E(φ | k2). Moreover, their complete integrals employ the parameterk2 as argument in place of the modulus k, i.e. K(k2) rather than K(k). And the integral of the third kind defined by Gradshteyn and Ryzhik, Π(φ, n, k), puts the amplitude φ first and not the "characteristic" n.
Thus one must be careful with the notation when using these functions, because various reputable references and software packages use different conventions in the definitions of the elliptic functions. For example, Wolfram's Mathematica software and Wolfram Alpha define the complete elliptic integral of the first kind in terms of the parameter m, instead of the elliptic modulus k.
Incomplete elliptic integral of the first kind
The incomplete elliptic integral of the first kindF is defined as
This is the trigonometric form of the integral; substituting t = sin θ and x = sin φ, one obtains the Legendre normal form:
Equivalently, in terms of the amplitude and modular angle one has:
With x = sn(u, k) one has: demonstrating that this Jacobian elliptic function is a simple inverse of the incomplete elliptic integral of the first kind.
The incomplete elliptic integral of the first kind has following addition theorem[citation needed]:
The elliptic modulus can be transformed that way:
Incomplete elliptic integral of the second kind
The incomplete elliptic integral of the second kindE in trigonometric form is
Substituting t = sin θ and x = sin φ, one obtains the Legendre normal form:
Equivalently, in terms of the amplitude and modular angle:
The incomplete elliptic integral of the second kind has following addition theorem[citation needed]:
The elliptic modulus can be transformed that way:
Incomplete elliptic integral of the third kind
The incomplete elliptic integral of the third kindΠ is
or
The number n is called the characteristic and can take on any value, independently of the other arguments. Note though that the value Π(1; π/2 | m) is infinite, for any m.
A relation with the Jacobian elliptic functions is
The meridian arc length from the equator to latitude φ is also related to a special case of Π:
Complete elliptic integral of the first kind
Plot of the complete elliptic integral of the first kind K(k)
Elliptic Integrals are said to be 'complete' when the amplitude φ = π/2 and therefore x = 1. The complete elliptic integral of the first kindK may thus be defined as or more compactly in terms of the incomplete integral of the first kind as
The complete elliptic integral of the first kind is sometimes called the quarter period. It can be computed very efficiently in terms of the arithmetic–geometric mean:[1]
Therefore the modulus can be transformed as:
This expression is valid for all and 0 ≤ k ≤ 1:
Relation to the gamma function
If k2 = λ(i√r) and (where λ is the modular lambda function), then K(k) is expressible in closed form in terms of the gamma function.[2] For example, r = 2, r = 3 and r = 7 give, respectively,[3]
and
and
More generally, the condition that be in an imaginary quadratic field[note 1] is sufficient.[4][5] For instance, if k = e5πi/6, then iK′/K = e2πi/3 and[6]
This approximation has a relative precision better than 3×10−4 for k < 1/2. Keeping only the first two terms is correct to 0.01 precision for k < 1/2.[citation needed]
Differential equation
The differential equation for the elliptic integral of the first kind is
A second solution to this equation is . This solution satisfies the relation
Continued fraction
A continued fraction expansion is:[7] where the nome is in its definition.
Inverse solution
Some applications require that k be calculated from a known ,[8] which we will refer to as for the remainder of this section, and the modulus of k, , will be referred to as k'. Although it is possible to obtain a value for k that solves the equation from table interpolations, Newton's method provides for a more efficient solution that is both rapid and precise for a wide variety of . Since the range of for 0.1<k<.9 is not too far off from linear, as observed from the table below, the standard equation for Newton's method is sufficient to solve the problem for this range. Limits to the states may be applied as needed to keep k within the range 0<k<1,
.
The extremely nonlinear nature of for k<0.1 and k>0.9 prevents the normal convergence of the equation in these ranges. However, when k is replaced by , and k' is replace by , then is significantly more linear in nature for high values, which in turn allows Newton's method to rapidly converge for F>.9. For F<.1, 1/F may be substituted for F, and the final k solution substituted with k'. The modified equations is
then
.
The derivatives in the denominators may be determined from the K(k) derivative definitions above and the rules from the derivative page. For high values of F, singularities occur due to finite 64 bit arithmetic precision. If it is not desired to use high precision arithmetic, approximations based on the limits as remove the singularities.
Then:
The initial value for lgk may be -F for the ln(k) case and k may be for the straight case, based on values observed in the table below. For smaller F, ln(1-tanh2(F)) and (1-tanh2(F)) have been observed to be a crude approximation and may serve as a suitable starting point.
Below is a table of , , , and the computed using Newton's method as described above.
1.e-20
-46.051702
30.199966
1.e-20
1.e-09
-20.723266
14.075383
1.e-09
1.e-05
-11.512925
8.2118984
1.e-05
0.001
-6.9077553
5.2801558
0.001
0.01
-4.6051702
3.8142689
0.01
0.1
-2.3025851
2.3468155
0.1
0.2
-1.6094379
1.9006702
0.2
0.5
-0.69314718
1.2792616
0.5
-0.34657359
1
0.70710678
0.8
-0.22314355
0.87743766
0.8
0.9
-0.10536052
0.72553432
0.9
0.99
-0.010050336
0.47032697
0.99
0.999
-0.0010005003
0.34958259
0.999
0.999999
-1.0000005E-06
0.1976472
0.999999
1-1.0e-09
-9.9999997E-10
0.1377727
1-1.0e-09
Complete elliptic integral of the second kind
Plot of the complete elliptic integral of the second kind E(k)
The complete elliptic integral of the second kindE is defined as
or more compactly in terms of the incomplete integral of the second kind E(φ,k) as
For an ellipse with semi-major axis a and semi-minor axis b and eccentricity e = √1 − b2/a2, the complete elliptic integral of the second kind E(e) is equal to one quarter of the circumferenceC of the ellipse measured in units of the semi-major axis a. In other words:
The complete elliptic integral of the second kind can be expressed as a power series[9]
which is equivalent to
In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as
The modulus can be transformed that way:
Computation
Like the integral of the first kind, the complete elliptic integral of the second kind can be computed very efficiently using the arithmetic–geometric mean.[1]
Define sequences an and gn, where a0 = 1, g0 = √1 − k2 = k′ and the recurrence relations an + 1 = an + gn/2, gn + 1 = √an gn hold. Furthermore, define
By definition,
Also
Then
In practice, the arithmetic-geometric mean would simply be computed up to some limit. This formula converges quadratically for all |k| ≤ 1. To speed up computation further, the relation cn + 1 = cn2/4an + 1 can be used.
Furthermore, if k2 = λ(i√r) and (where λ is the modular lambda function), then E(k) is expressible in closed form in terms of and hence can be computed without the need for the infinite summation term. For example, r = 1, r = 3 and r = 7 give, respectively,[10]
and
and
Derivative and differential equation
A second solution to this equation is E(√1 − k2) − K(√1 − k2).
Complete elliptic integral of the third kind
Plot of the complete elliptic integral of the third kind Π(n,k) with several fixed values of n
The complete elliptic integral of the third kindΠ can be defined as
Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the characteristicn,
Just like the complete elliptic integrals of the first and second kind, the complete elliptic integral of the third kind can be computed very efficiently using the arithmetic-geometric mean.[1]
Partial derivatives
Jacobi zeta function
In 1829, Jacobi defined the Jacobi zeta function: It is periodic in with minimal period . It is related to the Jacobi zn function by . In the literature (e.g. Whittaker and Watson (1927)), sometimes means Wikipedia's . Some authors (e.g. King (1924)) use for both Wikipedia's and .
Legendre's relation
The Legendre's relation or Legendre Identity shows the relation of the integrals K and E of an elliptic modulus and its anti-related counterpart[11][12] in an integral equation of second degree:
For two modules that are Pythagorean counterparts to each other, this relation is valid:
For example:
And for two modules that are tangential counterparts to each other, the following relationship is valid:
For example:
The Legendre's relation for tangential modular counterparts results directly from the Legendre's identity for Pythagorean modular counterparts by using the Landen modular transformation on the Pythagorean counter modulus.
Special identity for the lemniscatic case
For the lemniscatic case, the elliptic modulus or specific eccentricity ε is equal to half the square root of two. Legendre's identity for the lemniscatic case can be proved as follows:
According to the Chain rule these derivatives hold:
The Linear combination of the two now mentioned integrals leads to the following formula:
By forming the original antiderivative related to x from the function now shown using the Product rule this formula results:
If the value is inserted in this integral identity, then the following identity emerges:
This is how this lemniscatic excerpt from Legendre's identity appears:
Generalization for the overall case
Now the modular general case[13][14] is worked out. For this purpose, the derivatives of the complete elliptic integrals are derived after the modulus and then they are combined. And then the Legendre's identity balance is determined.
Because the derivative of the circle function is the negative product of the identical mapping function and the reciprocal of the circle function:
These are the derivatives of K and E shown in this article in the sections above:
In combination with the derivative of the circle function these derivatives are valid then:
Legendre's identity includes products of any two complete elliptic integrals. For the derivation of the function side from the equation scale of Legendre's identity, the Product rule is now applied in the following:
Of these three equations, adding the top two equations and subtracting the bottom equation gives this result:
In relation to the the equation balance constantly gives the value zero.
The previously determined result shall be combined with the Legendre equation to the modulus that is worked out in the section before:
The combination of the last two formulas gives the following result:
Because if the derivative of a continuous function constantly takes the value zero, then the concerned function is a constant function. This means that this function results in the same function value for each abscissa value and the associated function graph is therefore a horizontal straight line.
^Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 296
^Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 298
^Chowla, S.; Selberg, A. (1949). "On Epstein's Zeta Function (I)". Proceedings of the National Academy of Sciences. 35 (7): 373. Bibcode:1949PNAS...35..371C. doi:10.1073/PNAS.35.7.371. PMC 1063041. PMID 16588908. S2CID 45071481.
^Chowla, S.; Selberg, A. (1967). "On Epstein's Zeta-Function". Journal für die Reine und Angewandte Mathematik. 227: 86–110.
^N.Bagis,L.Glasser.(2015)"Evaluations of a Continued fraction of Ramanujan". Rend.Sem.Mat.Univ.Padova, Vol.133 pp 1-10
^Smith, John I. (May 5, 1971). "The Even- and Odd-Mode Capacitance Parameters for Coupled Lines in Suspended Substrate". IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. MTT-19 (5): 429 (Equation 37) – via IEEE Xplore.
^"Complete elliptic integral of the second kind: Series representations (Formula 08.01.06.0002)".
^Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7. p. 26, 161
^"Legendre-Relation" (in German). Retrieved 2022-11-29.
^"integration - Proving Legendres Relation for elliptic curves". Retrieved 2023-02-10.
^Internet Archive (1991), Paul Halmos celebrating 50 years of mathematics, New York : Springer-Verlag, ISBN 0-387-97509-8, retrieved 2023-02-10
Sources
Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 17". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 587. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
Byrd, P. F.; Friedman, M.D. (1971). Handbook of Elliptic Integrals for Engineers and Scientists (2nd ed.). New York: Springer-Verlag. ISBN 0-387-05318-2.
Carlson, B. C. (1995). "Numerical Computation of Real or Complex Elliptic Integrals". Numerical Algorithms. 10 (1): 13–26. arXiv:math/9409227. Bibcode:1995NuAlg..10...13C. doi:10.1007/BF02198293. S2CID 11580137.
Carlson, B. C. (2010), "Elliptic integral", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953). Higher transcendental functions. Vol II(PDF). McGraw-Hill Book Company, Inc., New York-Toronto-London. MR 0058756. Archived from the original (PDF) on 2011-07-14. Retrieved 2016-07-24.
Hancock, Harris (1910). Lectures on the Theory of Elliptic Functions. New York: J. Wiley & sons.
King, Louis V. (1924). On The Direct Numerical Calculation Of Elliptic Functions And Integrals. Cambridge University Press.
Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007), "Section 6.12. Elliptic Integrals and Jacobian Elliptic Functions", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8, archived from the original on 2011-08-11, retrieved 2011-08-09
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