# e-book Elliptic Functions and Elliptic Integrals

### 16.2 Functions and Variables for Elliptic Functions

Carlson, Numerical computation of real or complex elliptic integrals. Cox, The arithmetic-geometric mean of Gauss. Cremona, T. Enge, The complexity of class polynomial computation via floating point approximations. Enge, W. Hart, F. Johansson, Short addition sequences for theta functions. Integer Seq. Fieker, W. Hart, T.

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These integrals can be expressed in terms of the Jacobi zeta function defined in " Elliptic Integrals ". One of the most important properties of elliptic functions is that they are doubly periodic in the complex values of their arguments. Ordinary trigonometric functions are singly periodic, in the sense that for any integer. The elliptic functions are doubly periodic, so that for any pair of integers and. The Jacobi elliptic functions , etc.

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## 16. Elliptic Functions

Their periods include and , where is the complete elliptic integral of the first kind. The choice of "p" and "q" in the notation for Jacobi elliptic functions can be understood in terms of the values of the functions at the quarter periods and. The inverse function , for example, gives the value of for which. The inverse Jacobi elliptic functions are related to elliptic integrals. The four theta functions are obtained from EllipticTheta [ a , u , q ] by taking a to be 1 , 2 , 3 , or 4.

The functions are defined by , , ,. The theta functions are often written as with the parameter not explicitly given. The theta functions are sometimes written in the form , where is related to by. In addition, is sometimes replaced by , given by. It is defined by , where n runs over all p -dimensional integer vectors.

The Jacobi elliptic functions can be expressed as ratios of the theta functions. An alternative notation for theta functions is , , , , where. The Neville theta functions can be defined in terms of the theta functions as , , , , where. The Jacobi elliptic functions can be represented as ratios of the Neville theta functions. The Weierstrass function gives the value of for which. In order for any such value of to exist, and must be related by.

## Precise and Fast Computation of Elliptic Integrals and Functions - IEEE Conference Publication

The Weierstrass zeta and sigma functions are not strictly elliptic functions since they are not periodic. Modular elliptic functions are defined to be invariant under certain fractional linear transformations of their arguments. Thus for example is invariant under any combination of the transformations and. The definitions for elliptic integrals and functions given above are based on traditional usage. For modern algebraic geometry, it is convenient to use slightly more general definitions.

Integrals of the form can be expressed in terms of the ordinary logarithm and inverse trigonometric functions. You can think of EllipticLog as giving a generalization of this, where the polynomial under the square root is now of degree three.

It returns the list Null that appears in EllipticLog. EllipticExp is an elliptic function, doubly periodic in the complex plane. This quantity is central to many numerical algorithms for computing elliptic integrals and other functions. For positive reals and the AGM is obtained by starting with , , then iterating the transformation , until to the precision required. Enable JavaScript to interact with content and submit forms on Wolfram websites.

Learn how. Here is a plot of the complete elliptic integral of the second kind :. Here is with :.

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The elliptic integrals have a complicated structure in the complex plane:. Jacobi elliptic functions , etc.

## Elliptic Integrals

This shows two complete periods in each direction of the absolute value of the Jacobi elliptic function :. Give Feedback Top. Elliptic Integrals. Elliptic Modular Functions. Elliptic Functions. Generalized Elliptic Integrals and Functions.