Glossary

8.3 Elliptic trigonometric functions and identities

2 min read•july 22, 2024

Elliptic trigonometric functions are the non-Euclidean cousins of sine and cosine. They're defined on a curved plane, giving them unique properties like double periodicity and more complex graphs.

These functions play a key role in elliptic geometry, where triangles have angle sums greater than 180°. They're used in special versions of the law of cosines and sines for solving elliptic triangles.

Elliptic Trigonometric Functions

Definitions of elliptic trigonometric functions

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  • Analogous to Euclidean trigonometric functions but defined on the elliptic plane which is a non-Euclidean geometry with constant positive curvature
  • sn⁡(u,k)\operatorname{sn}(u, k) defined as the ratio of the elliptic y-coordinate to the elliptic radius
    • kk represents the elliptic modulus which determines the shape of the elliptic functions (e.g., k=0.5k = 0.5, k=0.8k = 0.8)
  • cn⁡(u,k)\operatorname{cn}(u, k) defined as the ratio of the elliptic x-coordinate to the elliptic radius
  • dn⁡(u,k)\operatorname{dn}(u, k) defined as the ratio of the elliptic radius to the semi-major axis of the ellipse

Fundamental identities in elliptic trigonometry

  • sn⁡2(u,k)+cn⁡2(u,k)=1\operatorname{sn}^2(u, k) + \operatorname{cn}^2(u, k) = 1 relates the elliptic sine and cosine functions
  • :
    1. sn⁡(u+v,k)=sn⁡(u,k)cn⁡(v,k)dn⁡(v,k)+sn⁡(v,k)cn⁡(u,k)dn⁡(u,k)1−k2sn⁡2(u,k)sn⁡2(v,k)\operatorname{sn}(u + v, k) = \frac{\operatorname{sn}(u, k)\operatorname{cn}(v, k)\operatorname{dn}(v, k) + \operatorname{sn}(v, k)\operatorname{cn}(u, k)\operatorname{dn}(u, k)}{1 - k^2\operatorname{sn}^2(u, k)\operatorname{sn}^2(v, k)}
    2. cn⁡(u+v,k)=cn⁡(u,k)cn⁡(v,k)−sn⁡(u,k)sn⁡(v,k)dn⁡(u,k)dn⁡(v,k)1−k2sn⁡2(u,k)sn⁡2(v,k)\operatorname{cn}(u + v, k) = \frac{\operatorname{cn}(u, k)\operatorname{cn}(v, k) - \operatorname{sn}(u, k)\operatorname{sn}(v, k)\operatorname{dn}(u, k)\operatorname{dn}(v, k)}{1 - k^2\operatorname{sn}^2(u, k)\operatorname{sn}^2(v, k)}
    3. dn⁡(u+v,k)=dn⁡(u,k)dn⁡(v,k)−k2sn⁡(u,k)sn⁡(v,k)cn⁡(u,k)cn⁡(v,k)1−k2sn⁡2(u,k)sn⁡2(v,k)\operatorname{dn}(u + v, k) = \frac{\operatorname{dn}(u, k)\operatorname{dn}(v, k) - k^2\operatorname{sn}(u, k)\operatorname{sn}(v, k)\operatorname{cn}(u, k)\operatorname{cn}(v, k)}{1 - k^2\operatorname{sn}^2(u, k)\operatorname{sn}^2(v, k)}

Comparing Elliptic and Euclidean Trigonometric Functions

Elliptic vs Euclidean trigonometric functions

  • Periodicity differs: elliptic functions have two periods that depend on the elliptic modulus kk, while Euclidean functions have one period
  • Range similarities: elliptic sine and cosine functions have a range of [-1, 1], like Euclidean functions
    • Elliptic delta amplitude function has a range of [1−k2\sqrt{1-k^2}, 1]
  • Graphs of elliptic functions are more complex and their shape depends on the value of the elliptic modulus kk (e.g., k=0.3k = 0.3, k=0.7k = 0.7)

Applications in elliptic geometry

  • Elliptic triangles have the sum of their angles greater than 180 degrees, unlike Euclidean triangles
  • Elliptic law of cosines relates the sides and angles of an elliptic triangle: cos⁡a=cos⁡A+cos⁡Bcos⁡Csin⁡Bsin⁡C\cos a = \frac{\cos A + \cos B \cos C}{\sin B \sin C}
    • aa, bb, and cc represent the sides and AA, BB, and CC are the corresponding opposite angles
  • Elliptic law of sines relates the sides and angles of an elliptic triangle: sin⁡Asin⁡a=sin⁡Bsin⁡b=sin⁡Csin⁡c\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}
  • Solving elliptic triangles involves using the elliptic laws of cosines and sines, along with the elliptic trigonometric functions and identities, to find unknown angles or sides (e.g., given two sides and an angle, find the remaining side and angles)

Key Terms to Review (19)

Abel's Theorem: Abel's Theorem is a fundamental result in the field of elliptic functions that establishes the relationship between the convergence of a series and the properties of its corresponding function. It plays a crucial role in understanding how elliptic functions behave and how they can be represented through identities. This theorem connects to various concepts such as the properties of elliptic integrals, the addition formulas for elliptic functions, and the structure of their function fields.
Analytic continuation: Analytic continuation is a technique in complex analysis used to extend the domain of a given analytic function beyond its original region of definition. This process allows for the evaluation of functions in new areas where the function may not be initially defined, maintaining the properties of the original function. The method is particularly significant when studying elliptic trigonometric functions, as it helps uncover deeper relationships and identities within their structures.
Carl Friedrich Gauss: Carl Friedrich Gauss was a prominent German mathematician and astronomer who made significant contributions to various fields, including number theory, statistics, and geometry. His work laid the groundwork for many concepts in non-Euclidean geometry and influenced the development of elliptic functions and spherical geometry.
Doubly Periodic: Doubly periodic refers to a property of functions that are periodic in two distinct directions, meaning they repeat their values in two different ways. This concept is particularly important in the study of elliptic functions, where the behavior of these functions is characterized by two fundamental periods that define a lattice in the complex plane, leading to rich mathematical structures and identities.
Elliptic angle addition formulas: Elliptic angle addition formulas are mathematical expressions that allow for the calculation of elliptic trigonometric functions for the sum of two angles. These formulas are crucial in understanding the relationships between elliptic functions and their corresponding angles, similar to traditional sine and cosine addition formulas in Euclidean geometry. They help in simplifying calculations involving elliptic integrals and play a significant role in non-Euclidean geometry, particularly in describing the geometry of surfaces with constant positive curvature.
Elliptic cosine function: The elliptic cosine function, denoted as $$ ext{cn}(u, k)$$, is a fundamental elliptic function that generalizes the standard cosine function to the context of elliptic curves. It is defined in terms of the complete elliptic integral and plays a crucial role in the study of elliptic functions, providing insights into the geometry of ellipses and non-Euclidean spaces.
Elliptic Curves: Elliptic curves are smooth, projective algebraic curves of genus one, equipped with a specified point, typically denoted as the 'point at infinity'. These curves have important applications in number theory, cryptography, and algebraic geometry. Their structure is defined by a cubic equation in two variables and is characterized by their group structure, allowing for the definition of elliptic functions and related identities.
Elliptic delta amplitude function: The elliptic delta amplitude function is a special function that arises in the context of elliptic integrals and elliptic functions, representing the inverse of the elliptic integral of the second kind. This function connects the properties of elliptic trigonometric functions, enabling calculations related to angles and arc lengths in non-Euclidean geometries, particularly in elliptic geometry. Its relationship with the elliptic sine and cosine functions is crucial for understanding the identities that govern these mathematical constructs.
Elliptic Integral: An elliptic integral is a type of integral that arises in the calculation of arc lengths of ellipses and is typically expressed in terms of elliptic functions. These integrals cannot generally be solved in terms of elementary functions, making them essential in the study of various mathematical and physical problems. They play a crucial role in connecting geometry with analysis, particularly when exploring the properties of elliptic curves and their applications.
Elliptic Pythagorean Identity: The elliptic Pythagorean identity is a fundamental relationship in elliptic trigonometry that expresses a connection between elliptic sine and elliptic cosine functions. Specifically, it states that for any angle, the sum of the squares of the elliptic sine and elliptic cosine functions equals one, much like the classical Pythagorean theorem in Euclidean geometry. This identity is crucial for understanding the properties and behaviors of elliptic trigonometric functions.
Elliptic Sine Function: The elliptic sine function, denoted as 'sn(u, k)', is a mathematical function that generalizes the ordinary sine function to the context of elliptic integrals and non-Euclidean geometry. This function is pivotal in defining elliptic trigonometric identities and provides a bridge between circular functions and their elliptic counterparts, particularly in describing the behavior of points on an elliptic curve.
Geodesic: A geodesic is the shortest path between two points on a curved surface, akin to a straight line in Euclidean geometry. In non-Euclidean geometries, geodesics play a crucial role as they define how distances and angles are perceived differently from flat surfaces, impacting various geometric properties and calculations.
Great Circle: A great circle is the largest possible circle that can be drawn on a sphere, resulting from the intersection of the sphere with a plane that passes through the center of the sphere. Great circles are fundamental in understanding various geometric properties on spheres, such as the shortest distance between two points, which connects them to concepts like area and excess in non-Euclidean settings and spherical trigonometry.
Jacobi elliptic functions: Jacobi elliptic functions are a set of basic functions that are periodic and are used to generalize trigonometric functions in the context of elliptic integrals. They provide a way to express the amplitude of a curve defined by an elliptic integral, relating them to the geometry of elliptic curves. These functions have applications across various fields, including physics, engineering, and number theory, making them a crucial concept when working with elliptic trigonometric functions and identities.
Modular forms: Modular forms are complex functions that are defined on the upper half-plane and exhibit certain transformation properties under the action of modular groups. They play a crucial role in number theory, especially in connection with elliptic curves, and can be seen as a bridge between geometry, analysis, and number theory.
Niels Henrik Abel: Niels Henrik Abel was a Norwegian mathematician known for his groundbreaking work in algebra and the development of elliptic functions. He made significant contributions that laid the foundation for elliptic trigonometric functions, which are essential in the study of geometry and analysis, particularly in non-Euclidean contexts. His work not only advanced the field of mathematics but also had a lasting impact on the development of modern mathematical theory.
Riemann's Period Relations: Riemann's period relations are fundamental equations that describe how the periods of elliptic functions are interrelated through complex analysis. These relations help establish connections between different elliptic integrals and their corresponding elliptic functions, ultimately leading to a deeper understanding of their geometric properties. They form a crucial part of the theory of Riemann surfaces and play a key role in the study of elliptic trigonometric functions and identities.
Theta function: Theta functions are special functions of several complex variables that arise in various areas of mathematics, particularly in the study of elliptic functions and the theory of modular forms. They can be used to express solutions to certain types of differential equations and are closely related to the geometry of Riemann surfaces, particularly in the context of elliptic curves and non-Euclidean geometry.
Weierstrass Elliptic Function: The Weierstrass elliptic function is a complex function that serves as a fundamental building block in the theory of elliptic functions, defined on the complex plane with two lattice points. These functions arise from the need to study periodic properties and can be expressed in terms of the Weierstrass $ ext{P}$ function, which is an even function with specific poles. This concept connects deeply with elliptic trigonometric functions and identities, allowing for a better understanding of their relationships and periodic behaviors.
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