Elliptic integral

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In calculation, one integral elliptical It's a function. $f$ form

{displaystyle f(x)=int _{c}^{x}Rleft(t,{sqrt {P(t)}}right)dt}

where $R$ is a rational function, $P$ is a polynomial without repeated roots and ${displaystyle cin mathbb {R} }$ .

The denomination elliptic integral starts from the first problems where these integrals took place, related to the calculation of the length of ellipse segments.

Elliptic integrals can be viewed as generalizations of inverse trigonometric functions. Elliptic integrals provide solutions to a somewhat broader class of problems than elementary inverse trigonometric functions, for example calculating the arc length of a circle only requires inverse trigonometric functions, but calculating the arc length of a ellipse requires elliptic integrals. Another good example is the pendulum, whose motion for small oscillations can be represented by trigonometric functions, but for larger oscillations requires the use of elliptic functions based on elliptic integrals.

Calculation

All elliptic integrals of the above type can be rewritten in terms of the sum of elementary functions and three "basic" types; of elliptic integrals (called first kind, second kind and third kind). To see this, let's write the elliptic integral in the form:

{displaystyle int {frac {P(w,x)}{Q(w,x)}} dx}

Where $w$ is a function of $x$ , such that ${displaystyle w^{2}}$ is a third or fourth grade polynomial, which contains at least an odd power $x$ .

Elliptic integral of the first kind

An elliptic integral of the first kind is a particular case of the elliptic integral. There are elliptic integrals of the first kind, complete and incomplete. The former depend on a single variable and the latter depend on two variables.

Complete elliptic integral of the first kind

The complete elliptical integral of first species $K$ is defined as:

{displaystyle K(x)=int _{0}^{pi /2}{frac {dtheta }{sqrt {1-x^{2}operatorname {sen} ^{2}theta }}}=int _{0}^{1}{frac {dv}{sqrt {(1-v^{2})(1-x^{2}v^{2})}}}}

and can be expressed as a power series as

{displaystyle K(x)={frac {pi }{2}}sum _{n=0}^{infty }left({frac {(2n)!}{2^{2n}(n!)^{2}}}right)^{2}x^{2n}={frac {pi }{2}}sum _{n=0}^{infty }left(P_{2n}(0)right)^{2}x^{2n}}

where ${displaystyle P_{n}}$ is the polynomial of Legendre, the previous expression is equivalent to

{displaystyle K(x)={frac {pi }{2}}left(1+left({frac {1}{2}}right)^{2}x^{2}+left({frac {1cdot 3}{2cdot 4}}right)^{2}x^{4}+cdots +left({frac {(2n-1)!!}{(2n)!!}}right)^{2}x^{2n}+cdots right)}

where ${displaystyle n!!}$ denotes double factorial.

Differential Equation

The differential equation for the elliptic integral of the first kind is

{displaystyle {frac {d}{dx}}left(x(1-x^{2}){frac {dK(x)}{dx}}right)=xK(x)}

A second solution for this equation is ${displaystyle Kleft(1-x^{2}right)}$ , this solution satisfies the relationship

{displaystyle {frac {d}{dx}}K(x)={frac {E(x)}{x(1-x^{2})}}-{frac {K(x)}{x}}}

Incomplete elliptic integral of the first kind

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

{displaystyle u=F(x,varphi)=int _{0}^{varphi }{frac {dtheta }{sqrt {1-x^{2}operatorname {sen} ^{2}theta }}}=int _{0}^{operatorname {sen} varphi }{frac {dv}{sqrt {(1-v^{2})(1-x^{2}v^{2})}}}=F_{x}(varphi)}

In this case the parameter ${displaystyle varphi =operatorname {am} (u)}$ it's called "amplitude" and if it's taken x as a parameter. This "ampness" is given by the reverse of the previous function F. The elliptical functions of Jacobi are defined from this breadth.

Landen transformation

The Landen transformation allows expressing incomplete elliptic integrals of one parameter into elliptic integrals of a different parameter. It can be proved that if we define a new amplitude φ₁ and a new parameter k₁, related to the old amplitude φ and the old parameter k by:

{displaystyle k_{1}={frac {2{sqrt {k}}}{1+k}}qquad tan varphi ={frac {operatorname {sen} 2varphi _{1}}{k+cos 2varphi _{1}}}}

Then there exists a simple relationship between the incomplete elliptic integrals associated with the parameters (k₁,φ₁) and ( k,φ) given by:

{displaystyle F(k,varphi)=int _{0}^{varphi }{frac {dtheta }{sqrt {1-k^{2}operatorname {sen} ^{2}theta }}}={frac {1}{1+k}}int _{0}^{varphi _{1}}{frac {dtheta _{1}}{sqrt {1-k_{1}^{2}operatorname {sen} ^{2}theta _{1}}}}}

This result can be applied iteratively to compute incomplete elliptic integrals in terms of elementary functions and limits. If we define the sequences:

{displaystyle k_{i}={frac {2{sqrt {k_{i+1}}}}{1+k_{i}}}qquad tan varphi _{i}={frac {operatorname {sen} 2varphi _{i+1}}{k+cos 2varphi _{i+1}}}}

So we have to:

{displaystyle F(k_{0},varphi _{0})={sqrt {frac {k_{1}k_{2}k_{3}dots }{k_{0}}}}int _{0}^{Phi }{frac {dtheta }{sqrt {1-operatorname {sen} ^{2}theta }}}={sqrt {frac {k_{1}k_{2}k_{3}dots }{k_{0}}}}ln tan left({frac {pi }{4}}+{frac {Phi }{2}}right)}

Where:

{displaystyle Phi =lim _{kto infty }varphi _{k}}

Elliptic integral of the second kind

An elliptic integral of the second kind is a particular case of the elliptic integral.

Complete elliptic integral of the second kind

The complete elliptical integral of second species $E$ is defined as:

{displaystyle E(x)=int _{0}^{pi /2}{sqrt {1-x^{2}operatorname {sen} ^{2}theta }} dtheta =int _{0}^{1}{frac {sqrt {1-x^{2}t^{2}}}{sqrt {1-t^{2}}}} dt}

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

{displaystyle E(x)={frac {pi }{2}}sum _{n=0}^{infty }left({frac {(2n)!}{2^{2n}(n!)^{2}}}right)^{2}{frac {x^{2n}}{1-2n}}}

equivalent to

{displaystyle E(x)={frac {pi }{2}}left(1-left({frac {1}{2}}right)^{2}x^{2}-left({frac {1cdot 3}{2cdot 4}}right)^{2}{frac {x^{4}}{3}}-left({frac {1cdot 3cdot 5}{2cdot 4cdot 6}}right)^{2}{frac {x^{6}}{5}}-dots left({frac {(2n-1)!!}{(2n)!!}}right)^{2}{frac {x^{2n}}{2n-1}}-dots right)}

Derivative and differential equation

{displaystyle {begin{aligned}&{frac {dE(x)}{dx}}={frac {E(x)-K(x)}{x}}\&(x^{2}-1){frac {d}{dx}}left(x{frac {dE(x)}{dx}}right)=xE(x)end{aligned}}}

Incomplete elliptic integral of the second kind

The incomplete elliptic integral of the second kind is a function of two variables that generalizes to the complete integral:

{displaystyle E(x,varphi)=int _{0}^{varphi }{sqrt {1-x^{2}operatorname {sen} ^{2}theta }};dtheta =int _{0}^{operatorname {sen} varphi }{frac {sqrt {1-x^{2}t^{2}}}{sqrt {1-t^{2}}}};dt}

Elliptic integral of the third kind

One elliptical integral of third species is a particular case of the integral elliptical. Sea $<math alttext="{displaystyle 0<k^{2}0.k2.1{displaystyle 0 ingredientk^{2}{2}{1}<img alt="{displaystyle 0<k^{2}$ , the complete elliptical integral of third species is defined as:

{displaystyle Pi (n,k)=int _{0}^{frac {pi }{2}}{dtheta over (1-noperatorname {sen} ^{2}theta){sqrt {1-k^{2}operatorname {sen} ^{2}theta }}}}

where $n$ It's a constant.

Applications

Elliptic integrals of the third kind appear naturally in the integration of the equations of motion of a spherical pendulum.

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