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The incomplete elliptic integral of the first kind F is defined as (,) = = (;) = .This is Legendre's trigonometric form of the elliptic integral; substituting t = sin θ and x = sin φ, one obtains Jacobi's algebraic form:
The incomplete elliptic integral of the first kind is defined as, (,) = (),the second kind as (,) = (),and the third kind as (,,) = ( ()) ().The argument n of the third kind of integral is known as the characteristic, which in different notational conventions can appear as either the first, second or third argument of Π and furthermore is sometimes defined with the opposite sign.
The incomplete elliptic integral of the first kind F is ... The integral may also be recognized as a multiple of Legendre's complete elliptic integral of the first kind.
In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms of the Carlson forms and vice versa.
where K and K′ are the complete elliptic integrals of the first kind for values satisfying k 2 + k′ 2 = 1, and E and E′ are the complete elliptic integrals of the second kind. This form of Legendre's relation expresses the fact that the Wronskian of the complete elliptic integrals (considered as solutions of a differential equation) is a ...
(the incomplete elliptic integral of the first kind) is computed. On the unit circle ( a = b = 1 {\displaystyle a=b=1} ), u {\displaystyle u} would be an arc length. However, the relation of u {\displaystyle u} to the arc length of an ellipse is more complicated.
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where = ( (/)) is the elliptic modulus and (,) is the incomplete elliptic integral of the second kind and sn, cn and dn are Jacobi's elliptic functions. [1] The surface of revolution is the nodoid constant mean curvature surface.