Elliptic function and integral
Armitage, J. Vernon Department of Mathematical Sciences, Science Laboratories, University of Durham, Durham, United Kingdom.
Erdélyi, A. Formerly, University of Edinburgh, Edinburgh, United Kingdom.
- Reduction of elliptic integrals
- Periods and singularities
- Inversion of elliptic integrals
- Doubly periodic functions
- Jacobian elliptic functions
- Weierstrass functions
- Theta functions
- Transformation theory
- Related Primary Literature
- Additional Reading
The elliptic functions were developed originally by K. F. Gauss, N. H. Abel, K. G. J. Jacobi, and others in order to solve the problem of finding the arc length of an ellipse, but they proved to have remarkable applications in physics, probability and statistics, algebra, geometry, and arithmetic. The elliptic functions generalize the circular (or trigonometric) functions (sin x, and so forth), which are restricted in their applications. The trigonometric functions are periodic in the sense that sin(x + 2π) = sin x, and so forth, where x is associated with the length of the arc of a circle. Elliptic functions are doubly periodic in the sense that there are two complex numbers called the periods. Those complex numbers are the vertices of a parallelogram, and we can picture that parallelogram as being folded over (using the periodicity to identify opposite sides) to obtain a torus. See also: Complex numbers and complex variables; Ellipse; Torus; Trigonometry
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