Elliptic function

In the mathematical branch of function theory , elliptic functions are double-period meromorphic functions . “Double period” means that there are two linearly independent periods in the real vector space in the form of two complex numbers , so that the two periodicity conditions are in the form of the functional equations ${\displaystyle \mathbb {C} }$ ${\displaystyle \omega _{1},\omega _{2}}$

${\displaystyle f(z+\omega _{1})=f(z)}$ and ${\displaystyle f(z+\omega _{2})=f(z)}$

for all are fulfilled. “Meromorph” means that the function is holomorphic everywhere except for isolated poles . H. there is infinitely differentiable and can be developed locally into a power series . ${\displaystyle z}$

The elliptic functions are inverse functions of the elliptic integrals .

Generalizations of the elliptic functions are the modular functions and the hyperelliptic functions .

Relation to ellipses

The name of the elliptic functions indicates that they were first used in calculating the circumference of ellipses . Another application is the calculation of the period of oscillation of a pendulum .

Periodic grid and basic mesh

If and, as above, are two independent complex numbers, the periods, then also applies ${\displaystyle f}$ ${\displaystyle \omega _{1},\omega _{2}}$

${\displaystyle f(z+\gamma )=f(z)}$

for any linear combination with integers , d. This means that nothing changes in the function value if you add an integer combination of them to the variable. The Abelian group ${\displaystyle \gamma =\mu \omega _{1}+\lambda \omega _{2}}$ ${\displaystyle \mu ,\lambda }$

${\displaystyle \Gamma =\langle \omega _{1},\omega _{2}\rangle _{\mathbb {Z} }=\mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}=\{\mu \omega _{1}+\lambda \omega _{2}\mid \mu ,\lambda \in \mathbb {Z} \}}$

is called the periodic grid . It's a full grid in . ${\displaystyle \mathbb {C} }$

The parallelogram spanned by and ${\displaystyle \omega _{1}}$ ${\displaystyle \omega _{2}}$

${\displaystyle \{\mu \omega _{1}+\lambda \omega _{2}\mid 0\leq \mu ,\lambda \leq 1\}}$

is called the basic mesh of the grid.

Interpreted geometrically, this linear superposition of the periods is a tiling of the complex plane with parallelograms. Everything that happens in one parallelogram is repeated in every other. If opposite sides of the parallelogram are identified, this corresponds topologically to a torus . Just as the trigonometric functions sine and cosine are related to the circle, elliptic functions belong to the torus.

Simple properties

• A holomorphic elliptic function is constant: it is whole , and it is bounded, since it already takes all of its values ​​on the basic mesh and the basic mesh is compact . According to Liouville's theorem , it is therefore constant.
• An elliptic function cannot have exactly one single pole in the basic mesh, since the integral over the edge of the basic mesh vanishes due to the periodicity; according to Cauchy's integral formula, this excludes a simple pole.
• The corresponding statement also applies to exactly one simple zero, since it is also an elliptic function. ${\displaystyle f}$ ${\displaystyle 1/f}$

The Weierstrasse function ${\displaystyle \wp }$

For a period lattice there is always a non-constant elliptic function, the Weierstrasse sche-function (the symbol is called Weierstrasse-p ): ${\displaystyle \Gamma }$ ${\displaystyle \wp }$

${\displaystyle \wp (z)={\frac {1}{z^{2}}}+\sum _{\gamma \in \Gamma \setminus \{0\}}\left({\frac {1}{(z-\gamma )^{2}}}-{\frac {1}{\gamma ^{2}}}\right).}$

Essentially, translation is used to make an -invariant function; the summands only serve to make the series convergent. ${\displaystyle 1/z^{2}}$ ${\displaystyle \Gamma }$ ${\displaystyle -1/\gamma ^{2}}$

${\displaystyle \wp }$is an even elliptic function; H. . Your derivation ${\displaystyle \wp (-z)=\wp (z)}$

${\displaystyle \wp '(z)=-2\sum _{\gamma \in \Gamma }{\frac {1}{(z-\gamma )^{3}}}}$

is an odd elliptic function; H. ${\displaystyle \wp '(-z)=-\wp '(z).}$

The central result of the theory of elliptic functions is the following statement: Every elliptic function for the period lattice can be written as a rational function in and . Every relation between and follows from the differential equation of the function ${\displaystyle \Gamma }$ ${\displaystyle \wp }$ ${\displaystyle \wp '}$ ${\displaystyle \wp }$ ${\displaystyle \wp '}$ ${\displaystyle \wp }$

${\displaystyle (\wp '(z))^{2}=4\wp (z)^{3}-g_{2}(\Gamma )\wp (z)-g_{3}(\Gamma ).}$

There are constants that depend on, are more precise and Eisenstein rows to the lattice . In algebraic language this sentence means: The body of the elliptical functions to the period lattice is isomorphic to the body ${\displaystyle g_{2}(\Gamma ),g_{3}(\Gamma )}$ ${\displaystyle \Gamma }$ ${\displaystyle g_{2}(\Gamma )=60G_{4}(\Gamma )}$ ${\displaystyle g_{3}(\Gamma )=140G_{6}(\Gamma )}$ ${\displaystyle \Gamma }$ ${\displaystyle \Gamma }$

${\displaystyle \mathbb {C} (X)[Y]/(Y^{2}-4X^{3}+g_{2}X+g_{3}).}$

This isomorphism shows up and down . ${\displaystyle \wp }$ ${\displaystyle X}$ ${\displaystyle \wp '}$ ${\displaystyle Y}$

• 

  Weierstrass ℘ function for the grid in the area . The zeros appear black and the poles white. ${\displaystyle \Gamma =[1,e^{\pi i/3}]}$ ${\displaystyle [-2,2]\times [-1{,}125,1{,}125]i}$

• 

  Derivation of this ℘-function in the same area and with the same coloring.

• 

  ℘ function for the grid in the same area and with the same coloring. ${\displaystyle \Gamma =[1,{\tfrac {1}{2}}+{\tfrac {i}{3}}]}$

On the history of elliptical functions

This area was founded soon after the development of calculus by the Italian mathematician Giulio di Fagnano (1682–1766) and the Swiss mathematician Leonhard Euler (1707–1783). When calculating the arc length of an ellipse, they encountered problems that involved integrals in which the square roots of 3rd and 4th degree polynomials appeared. It was recognized that they could not be expressed in a closed form using the functions that had been in use until then. However, Fagnano noticed a relationship between the arc lengths of various special arcs that he published in 1750. Euler encountered the same relationship in connection with the same problem and brought them into connection with one another in 1766 with the aid of a theorem according to which he could represent the sum of certain such integrals as an integral of the same kind. He emphasized that these integrals, like the cyclometric functions and the logarithmic function , could be introduced into mathematics as symbols.

Apart from a remark by Landen  , his ideas were only pursued further in 1786 by Legendre (1752-1833) in his two Mémoires sur les intégrations par arcs d'ellipse (treatises on integration through elliptical arcs ). From then on, Legendre has repeatedly dealt with this type of integrals and called them "elliptic functions". Legendre's works should also be mentioned: Mémoire sur les transcendantes elliptiques (1792), Exercices de calcul intégral (1811–1817), Traité des fonctions elliptiques (1825–1832). Legendre traced the elliptical functions back to three fixed forms - genera - which made it much easier for himself to study them, which was very difficult at the time. However, his work went completely unnoticed until 1826.

It was only from then on that the two mathematicians Abel (1802–1829) and Jacobi (1804–1851) resumed these investigations and quickly came to unexpected new discoveries. First, they reversed the problem by considering the imaginary upper limit of the integral as a function of the integral value, i.e. considering the functions that are inverse to the elliptic integrals. These inverse functions are now called elliptic functions according to a proposal by Jacobi from 1829 . The works of Jacobi and Abel can be found in Crelle's Journal of 1826. Jacobi's Fundamenta nova theoriae functionum ellipticarum (1829) should also be mentioned. Jacobi proved in 1835 that the unique functions of a variable have at most two independent periods. The elliptic functions have exactly two. The addition theorem, which Euler found in a very special form, was expressed and proven in its general form by Abel in 1829. Gauss , as he himself noted and as has been proven, had already found many of the properties of the elliptical functions thirty years earlier, but had not published anything about them.

Further development has led to the hyperelliptic functions (the Abelian functions ) and the module functions .

See also

• Elliptic curve
• Jacobian elliptic function
• Lemniscatic sine and cosine functions
• Theta function
• Rational elliptic functions

literature

• Heinrich Burkhardt: Elliptical functions. 3. Edition. Association of Scientific Publishers, Berlin [u. a.] 1920. (Lectures on Function Theory, Volume 2.)
• Heinrich Durège, Ludwig Maurer: Theory of the elliptical functions. 5th edition. Teubner, Leipzig 1908.
• Eberhard Freitag, Rolf Busam: Function theory 1. 4. Edition. Springer, Berlin [a. a.] 2006, ISBN 3-540-31764-3 .
• Robert Fricke : The elliptical functions and their applications. 3 volumes. (Volume 1, Volume 2 , Volume 3 (2011) published posthumously). Teubner, Berlin / Leipzig 1916–1922, 2nd edition 1930. ND Springer, Berlin / Heidelberg [u. a.] 2011, ISBN 978-3-642-19556-3 , ISBN 978-3-642-19560-0 , ISBN 978-3-642-20953-6 .
• Adolf Hurwitz, Richard Courant: Lectures on general function theory and elliptic functions. 4th edition. Springer, Berlin [a. a.] 1964. (The basic teachings of the mathematical sciences in individual representations, Vol. 3.) 5th edition. Springer, Berlin / Heidelberg [a. a.] 2000.
• Max Koecher , Aloys Krieg : Elliptical functions and modular forms. 2nd Edition. Springer, Berlin [a. a.] 2007, ISBN 978-3-540-49324-2 .
• Francesco Giacomo Tricomi, Maximilian Krafft: Elliptical functions. Akademische Verlagsgesellschaft, Leipzig 1948 (Mathematics and its Applications in Physics and Technology, Series A, Volume 20.)

Individual evidence

1. ↑ John Landen: An Investigation of a general Theorem for finding the Length of any Arc of any Conic Hyperbola, by Means of Two Elliptic Arcs, with some other new and useful Theorems deduced therefrom. In: The Philosophical Transactions of the Royal Society of London 65 (1775), No. XXVI, pp. 283-289, JSTOR 106197 .
2. ^ Adrien-Marie Legendre: Mémoire sur les intégrations par arcs d'ellipse. In: Histoire de l'Académie royale des sciences Paris (1788), pp. 616–643. - Ders .: Second mémoire sur les intégrations par arcs d'ellipse, et sur la comparaison de ces arcs. In: Histoire de l'Académie royale des sciences Paris (1788), pp. 644–683.
3. ^ Adrien-Marie Legendre: Mémoire sur les transcendantes elliptiques , ou l'on donne des méthodes faciles pour comparer et évaluer ces trancendantes, qui comprennent les arcs d'ellipse, et qui se rencontrent frèquemment dans les applications du calcul intégral. Du Pont & Firmin-Didot, Paris 1792. English translation A Memoire on Elliptic Transcendentals. In: Thomas Leybourn: New Series of the Mathematical Repository . Volume 2. Glendinning, London 1809, Part 3, pp. 1-34.
4. ^ Adrien-Marie Legendre: Exercices de calcul integral sur divers ordres de transcendantes et sur les quadratures. 3 volumes. ( Volume 1 , Volume 2 , Volume 3). Paris 1811-1817.
5. ^ Adrien-Marie Legendre: Traité des fonctions elliptiques et des intégrales eulériennes, avec des tables pour en faciliter le calcul numérique. 3 vols. ( Volume 1 , Volume 2 , Volume 3/1 , Volume 3/2, Volume 3/3). Huzard-Courcier, Paris 1825–1832.
6. ^ Carl Gustav Jacob Jacobi: Fundamenta nova theoriae functionum ellipticarum. Koenigsberg 1829.