Elliptic integrals

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 (). 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 which can be expressed in the form

$f(x) = \int_^ R \left(t, \sqrt \right) \, dt,$

where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant.

In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of 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:
* , the ''modular angle''
* , the ''elliptic modulus'' or ''eccentricity''
* , 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 , the ''amplitude'', or as or , where and is one of the Jacobian elliptic functions.

Specifying the value of any one of these quantities determines the others. Note that also depends on . Some additional relationships involving include
$\cos \varphi = \operatorname u, \quad \textrm \quad \sqrt = \operatorname u.$

The latter is sometimes called the ''delta amplitude'' and written as . 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:
$F(\varphi, \sin \alpha) = F\left(\varphi \mid \sin^2 \alpha\right) = F(\varphi \setminus \alpha) = F(\sin \varphi ; \sin \alpha).$
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, , is often encountered; and similarly for the integral of the second kind. Abramowitz and Stegun substitute the integral of the first kind, , for the argument in their definition of the integrals of the second and third kinds, unless this argument is followed by a vertical bar: i.e. for . Moreover, their complete integrals employ the ''parameter'' as argument in place of the modulus , i.e. rather than . And the integral of the third kind defined by Gradshteyn and Ryzhik, , puts the amplitude first and not the "characteristic" .

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 , instead of the elliptic modulus .

Incomplete elliptic integral of the first kind

The incomplete elliptic integral of the first kind is defined as

$F(\varphi,k) = F\left(\varphi \mid k^2\right) = F(\sin \varphi ; k) = \int_0^\varphi \frac .$

This is the trigonometric form of the integral; substituting and , one obtains the Legendre normal form:

$F(x ; k) = \int_^ \frac.$

Equivalently, in terms of the amplitude and modular angle one has:
$F(\varphi \setminus \alpha) = F(\varphi, \sin \alpha) = \int_0^\varphi \frac.$

With one has:
$F(x;k) = u;$
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:
F\bigl arctan(x),k\bigr+ F\bigl arctan(y),k\bigr= F\left arctan\left(\frac\right) + \arctan\left(\frac\right),k\right

The elliptic modulus can be transformed that way:
F\bigl arcsin(x),k\bigr= \fracF\left arcsin\left(\frac\right),\frac\right

Incomplete elliptic integral of the second kind

The incomplete elliptic integral of the second kind in trigonometric form is

$E(\varphi,k) = E\left(\varphi \,, \,k^2\right) = E(\sin\varphi;k) = \int_0^\varphi \sqrt\, d\theta.$

Substituting and , one obtains the Legendre normal form:

$E(x;k) = \int_0^x \frac\,dt.$

Equivalently, in terms of the amplitude and modular angle:
$E(\varphi \setminus \alpha) = E(\varphi, \sin \alpha) = \int_0^\varphi \sqrt \, d\theta.$

Relations with the Jacobi elliptic functions include
$\begin E\bigl(\operatorname(u ; k) ; k\bigr) &= \int_0^u \operatorname^2 (w ; k) \,dw \\ &= u - k^2 \int_0^u \operatorname^2 (w ; k) \,dw \\ &= \left(1-k^2\right)u + k^2 \int_0^u \operatorname^2 (w ; k) \,dw. \end$

The meridian arc length from the equator to latitude is written in terms of :
$m(\varphi) = a\left(E(\varphi,e)+\fracE(\varphi,e)\right),$
where is the semi-major axis, and is the eccentricity.

The incomplete elliptic integral of the second kind has following addition theorem:
E\bigl arctan(x),k\bigr+ E\bigl arctan(y),k\bigr= E\left arctan\left(\frac\right) + \arctan\left(\frac\right),k\right+ \frac\left(\frac+\frac\right)

The elliptic modulus can be transformed that way:
E\bigl arcsin(x),k\bigr= \left(1+\sqrt\right)E\left arcsin\left(\frac\right),\frac\right- \sqrtF\bigl arcsin(x),k\bigr+ \frac

Incomplete elliptic integral of the third kind

The incomplete elliptic integral of the third kind is
$\Pi(n ; \varphi \setminus \alpha) = \int_0^\varphi \frac \frac$

or

$\Pi(n ; \varphi \,, \,m) = \int_^ \frac \frac.$

The number is called the characteristic and can take on any value, independently of the other arguments. Note though that the value is infinite, for any .

A relation with the Jacobian elliptic functions is
$\Pi\bigl(n; \,\operatorname(u;k); \,k\bigr) = \int_0^u \frac .$

The meridian arc length from the equator to latitude is also related to a special case of :

$m(\varphi)=a\left(1-e^2\right)\Pi\left(e^2 ; \varphi \,, \,e^2\right).$

Complete elliptic integral of the first kind

Elliptic Integrals are said to be 'complete' when the amplitude and therefore . The complete elliptic integral of the first kind may thus be defined as
$K(k) = \int_0^\tfrac \frac = \int_0^1 \frac,$
or more compactly in terms of the incomplete integral of the first kind as
$K(k) = F\left(\tfrac,k\right) = F\left(\tfrac \,, \, k^2\right) = F(1;k).$

It can be expressed as a power series
$K(k) = \frac\sum_^\infty \left(\frac\right)^2 k^ = \frac \sum_^\infty \bigl(P_(0)\bigr)^2 k^,$

where is the Legendre polynomials, which is equivalent to

$K(k) = \frac\left(1+\left(\frac\right)^2 k^2+\left(\frac\right)^2 k^4+\cdots+\left(\frac\right)^2 k^+\cdots\right),$

where denotes the double factorial. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as

$K(k) = \tfrac \,_2F_1 \left(\tfrac, \tfrac; 1; k^2\right).$

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:
$K(k) = \frac.$

Therefore the modulus can be transformed that way:
\begin
K(k) &= \frac \\ pt& = \frac \\ pt&= \frac \\ pt& = \fracK\left(\frac\right)
\end

This expression is valid for all $n \isin \mathbb$ and :
K(k) = n\left sum_^ \operatorname\left(\fracK(k);k\right)\rightK\left ^n\prod_^\operatorname\left(\fracK(k);k\right)^2\right

Relation to the gamma function

If and $r \isin \mathbb^+$ (where is the modular lambda function), then is expressible in closed form in terms of the gamma function. For example, and give, respectively,

$K\left(\sqrt-1\right)=\frac,$

and

$K\left(\frac\right)=\frac.$

More generally, the condition that
$\frac=\frac$
be in an imaginary quadratic field can be analytically extended to the complex plane. is sufficient. For instance, if , then and

$K\left(e^\right)=\frac.$

Relation to Jacobi theta function

The relation to Jacobi's theta function is given by
$K(k)=\frac\theta_3^2(q),$
where the nome is
$q(k) = \exp\left(-\pi \frac\right).$

Asymptotic expressions

$K\left(k\right)\approx\frac+\frac\frac-\frac\frac$
This approximation has a relative precision better than for . Keeping only the first two terms is correct to 0.01 precision for .

Differential equation

The differential equation for the elliptic integral of the first kind is
$\frac\left(k\left(1-k^2\right)\frac\right) = k \, K(k)$

A second solution to this equation is . This solution satisfies the relation
$\fracK(k) = \frac-\frac.$

Continued fraction

A continued fraction expansion is:
$\frac = -\frac + \sum^_ \frac = -\frac + \cfrac,$
where the nome is .

Complete elliptic integral of the second kind

The complete elliptic integral of the second kind is defined as

$E(k) = \int_0^\tfrac \sqrt \, d\theta = \int_0^1 \frac \, dt,$

or more compactly in terms of the incomplete integral of the second kind as

$E(k) = E\left(\tfrac,k\right) = E(1;k).$

For an ellipse with semi-major axis and semi-minor axis and eccentricity , the complete elliptic integral of the second kind is equal to one quarter of the circumference of the ellipse measured in units of the semi-major axis . In other words:

$C = 4 a E(e).$

The complete elliptic integral of the second kind can be expressed as a power series

$E(k) = \frac\sum_^\infty \left(\frac\right)^2 \frac,$

which is equivalent to

$E(k) = \frac\left(1-\left(\frac12\right)^2 \frac-\left(\frac\right)^2 \frac-\cdots-\left(\frac\right)^2 \frac-\cdots\right).$

In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as

$E(k) = \tfrac \,_2F_1 \left(\tfrac12, -\tfrac12; 1; k^2 \right).$

The modulus can be transformed that way:
$E(k) = \left(1+\sqrt\right)\,E\left(\frac\right) - \sqrt\,K(k)$

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.

Define sequences and , where , and the recurrence relations , hold. Furthermore, define
$c_n=\sqrt.$

By definition,

$a_\infty = \lim_ a_n = \lim_ g_n = \operatorname\left(1, \sqrt\right).$

Also

$\lim_ c_n=0.$

Then

$E(k) = \frac\left(1-\sum_^ 2^ c_n^2\right).$

In practice, the arithmetic-geometric mean would simply be computed up to some limit. This formula converges quadratically for all . To speed up computation further, the relation can be used.

Furthermore, if and $r \isin \mathbb^+$ (where is the modular lambda function), then is expressible in closed form in terms of
$K(k)=\frac$
and hence can be computed without the need for the infinite summation term. For example, and give, respectively, p. 26, 161

$E\left(\frac\right)=K\left(\frac\right)+\frac,$
and
$E\left(\frac\right)=\fracK\left(\frac\right)+\frac.$

Derivative and differential equation

$\frac = \frac$
$\left(k^2-1\right) \frac \left( k \;\frac \right) = k E(k)$

A second solution to this equation is .

Complete elliptic integral of the third kind

The complete elliptic integral of the third kind can be defined as

$\Pi(n,k) = \int_0^\frac \frac.$

Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the ''characteristic'' ,
$\Pi'(n,k) = \int_0^\frac \frac.$

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.

Partial derivatives

\begin
\frac &= \frac\left(E(k)+\frac\left(k^2-n\right)K(k) + \frac \left(n^2-k^2\right)\Pi(n,k)\right) \\ pt
\frac &= \frac\left(\frac+\Pi(n,k)\right)
\end

Functional relations

Legendre's relation:
$K(k) E\left(\sqrt\right) + E(k) K\left(\sqrt\right) - K(k) K\left(\sqrt\right) = \frac \pi 2.$

See also

* Elliptic curve
* Schwarz–Christoffel mapping
* Carlson symmetric form
* Jacobi's elliptic functions
* Weierstrass's elliptic functions
* Jacobi theta function
* Ramanujan theta function
* Arithmetic–geometric mean
* Pendulum period
* Meridian arc

References

Notes

References

Sources

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External links

*
Eric W. Weisstein, "Elliptic Integral" (Mathworld)Matlab code for elliptic integrals evaluation
by elliptic project

(Exstrom Laboratories)
A Brief History of Elliptic Integral Addition Theorems

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Elliptic functions
Special hypergeometric functions