What are elliptical integrals used for?
Elliptic integrals can be viewed as generalizations of the inverse trigonometric functions and provide solutions to a wider class of problems. For instance, while the arc length of a circle is given as a simple function of the parameter, computing the arc length of an ellipse requires an elliptic integral.
How do you find the integral of an elliptic?
The complete elliptic integral is obtained by setting the amplitude φ = π/2 or sin φ = 1, the maximum range on the upper bound of integration for the elliptic integral.
What is elliptic integral of first kind?
The complete elliptic integral of the first kind K(k) is defined for 0<k<1 0 < k < 1 by K(k):=∫π20dθ√1−k2sin2θ. … The real number k is called the modulus of the elliptic integral. The complementary modulus is k′=(1−k2)12 k ′ = ( 1 − k 2 ) 1 2 (0<k′<1 0 < k ′ < 1 ).
Why is elliptic functions important?
In addition to their use in applied mathematics, the development of the theory of elliptic functions also spurred the study of functions of complex variables and provided a bridge between pure and applied mathematics.
What is line integral in mathematics?
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.
Who discovered elliptic integrals?
Elliptic integrals were intensively studied for many years by the French mathematician Adrien-Marie Legendre, who was able to calculate tables of values for such expressions as functions of their upper endpoint, x. But the topic was completely transformed in the late 1820s by the independent…
What is incomplete elliptic integral of the second kind?
Incomplete Elliptic Integral of the Second Kind Note that some definitions use the elliptical modulus k or the modular angle α instead of the parameter m. They are related as m = k2 = sin2α.
Who discovered elliptic functions?
At the beginning of the 19th century, elliptic functions were discovered independently and almost simultaneously by Abel and Jacobi. The theta functions mentioned above in Section 5.15 and extensively studied by Jacobi were an essential tool in his work on elliptic functions.