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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.
Landen's transformation is a mapping of the parameters of an elliptic integral, useful for the efficient numerical evaluation of elliptic functions. It was originally due to John Landen and independently rediscovered by Carl Friedrich Gauss .
The duplication theorem can be used for a fast and robust evaluation of the Carlson symmetric form of elliptic integrals and therefore also for the evaluation of Legendre-form of elliptic integrals. Let us calculate R F ( x , y , z ) {\displaystyle R_{F}(x,y,z)} : first, define x 0 = x {\displaystyle x_{0}=x} , y 0 = y {\displaystyle y_{0}=y ...
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.
Plot of the hypergeometric function 2F1(a,b; c; z) with a=2 and b=3 and c=4 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D In mathematics , the Gaussian or ordinary hypergeometric function 2 F 1 ( a , b ; c ; z ) is a special function represented by the hypergeometric series , that ...
That web page gives the incomplete elliptic integrals, not the complete elliptic integrals. I'd like if someone who was good with all this make the notation section better at explaining the problems with notation people have with these squares. 09:26, 30 September 2011 (UTC)