Many other problems in electromagnetism and gravitation are solved by elliptic integrals
You may look at the excersices in the bookleast_action said:On the page about http://mathworld.wolfram.com/EllipticIntegral.html" it says:
Where could I find a collection of such problems? or do you have any nice examples of this?
Elliptic integrals are a class of special functions that arise in the evaluation of integrals involving elliptic curves. They are important in science because they have numerous applications in physics, engineering, and mathematics, including in the fields of celestial mechanics, electromagnetism, and number theory.
Elliptic integrals are different from other types of integrals in that they involve integrands containing square roots of quartic or higher order polynomials. In other words, they cannot be expressed in terms of elementary functions and require special functions, such as elliptic integrals, to be evaluated.
Some real-world applications of elliptic integrals include calculating the period of a pendulum, determining the trajectory of a planet orbiting the sun, and solving problems in electromagnetism. They are also used in the design of electrical circuits, antennas, and satellite orbits.
Yes, there are three types of elliptic integrals: elliptic integrals of the first kind, which involve integrands with no square roots; elliptic integrals of the second kind, which involve integrands with one square root; and elliptic integrals of the third kind, which involve integrands with two or more square roots.
Elliptic integrals are closely related to elliptic curves, which are curves described by equations of the form y^2 = x^3 + ax + b. In fact, the inverse of an elliptic integral can be used to parameterize an elliptic curve. This connection has led to many important applications of elliptic curves in cryptography, number theory, and other areas of mathematics.