# Inversion of incomplete elliptic integral of the second kind

• xdrgnh
In summary: To invert this integral symbolically, one can use a coordinate transformation to convert it to an elliptic integral of the first kind, for which the inverse function is known. This method would be preferable over numerical inversion, which may not yield accurate results.

#### xdrgnh

Hello I hope this is the right place to ask this question. For my thesis I need a way to invert a incomplete elliptic integral of the second kind. I believe the Jacobi elliptic functions are inverse of the elliptic integral of the first kind. The calculation I'm doing is symbolic so a numerically inverting the second integral will be no good for me. Does anyone know which function is the inverse to the elliptic integral of the second. Or perhaps I can do a coordinate transformation to turn my elliptic integral of the second kind to a elliptic integral of the first kind.

xdrgnh said:
Hello I hope this is the right place to ask this question. For my thesis I need a way to invert a incomplete elliptic integral of the second kind. I believe the Jacobi elliptic functions are inverse of the elliptic integral of the first kind. The calculation I'm doing is symbolic so a numerically inverting the second integral will be no good for me. Does anyone know which function is the inverse to the elliptic integral of the second. Or perhaps I can do a coordinate transformation to turn my elliptic integral of the second kind to a elliptic integral of the first kind.

Incomplete elliptic integrals of the second kind can be expressed in terms of Jacobi elliptic functions:

http://en.wikipedia.org/wiki/Elliptic_integral#Incomplete_elliptic_integral_of_the_second_kind

## 1. What is an incomplete elliptic integral of the second kind?

An incomplete elliptic integral of the second kind is a mathematical function that is used to calculate the arc length of an ellipse. It is denoted by the symbol E(phi, m), where phi is the amplitude and m is the elliptic modulus.

## 2. What is the significance of inverting the incomplete elliptic integral of the second kind?

Inverting the incomplete elliptic integral of the second kind means finding the values of the amplitude and elliptic modulus when the arc length is known. This is useful in solving various problems in physics, engineering, and other fields.

## 3. How is the incomplete elliptic integral of the second kind calculated?

The incomplete elliptic integral of the second kind can be calculated using various methods, such as series expansion, numerical integration, and special functions. The specific method used depends on the values of the amplitude and elliptic modulus.

## 4. What are the applications of the incomplete elliptic integral of the second kind?

The incomplete elliptic integral of the second kind has various applications in mathematics, physics, and engineering. It is used in calculating the period of a pendulum, solving problems involving motion on an elliptic curve, and in the design of electronic filters.

## 5. Are there any real-life examples of the incomplete elliptic integral of the second kind?

Yes, the incomplete elliptic integral of the second kind can be seen in various real-life situations. For example, it is used in calculating the trajectory of a satellite orbiting around a planet, determining the shape of a planetary orbit, and in the design of curved mirrors in telescopes.

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