Explicit expressions for the quarter periods, in terms of the nome, are given in the linked article.
They are defined as follows: The nome solves the following equation: This analogon is valid for the Pythagorean complementary modulus: where
For the complete theta functions these definitions introduced by Sir Edmund Taylor Whittaker and George Neville Watson are valid: These three definition formulas are written down in the fourth edition of the book A Course in Modern Analysis written by Whittaker and Watson on the pages 469 and 470.
The nome is commonly used as the starting point for the construction of Lambert series, the q-series and more generally the q-analogs.
of q-series then arises in the theory of affine Lie algebras, essentially because (to put it poetically, but not factually)[citation needed] those algebras describe the symmetries and isometries of Riemann surfaces.
The elliptic nome function is axial symmetric to the ordinate axis.
The functional curve of the nome passes through the origin of coordinates with the slope zero and curvature plus one eighth.
The Legendre's relation is defined that way: And as described above, the elliptic nome function
And the integer number sequence in MacLaurin series of that elliptic period ratio leads to the integer sequence of the series of the elliptic nome directly.
The German mathematician Adolf Kneser researched on the integer sequence of the elliptic period ratio in his essay Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen and showed that the generating function of this sequence is an elliptic function.
Also a further mathematician with the name Robert Fricke analyzed this integer sequence in his essay Die elliptischen Funktionen und ihre Anwendungen and described the accurate computing methods by using this mentioned sequence.
leads to this equation that shows the generating function of the Kneser number sequence: This result appears because of the Legendre's relation
The mathematician Karl Heinrich Schellbach [de] discovered the integer number sequence that appears in the MacLaurin series of the fourth root of the quotient Elliptic Nome function divided by the square function.
[1]: 60 The sequence was also constructed by the Silesian German mathematician Hermann Amandus Schwarz in Formeln und Lehrsätze zum Gebrauche der elliptischen Funktionen[2] (pages 54–56, chapter Berechnung der Grösse k).
This Schellbach Schwarz number sequence Sc(n) was also analyzed by the mathematicians Karl Theodor Wilhelm Weierstrass and Louis Melville Milne-Thomson in the 20th century.
was researched by the Czech mathematician and fairy chess composer Václav Kotěšovec, born in 1956.
The Kotěšovec numbers are generated in the same way as the Schellbach Schwarz numbers are constructed: The only difference consists in the fact that this time the factor before the sum in this corresponding analogous formula is not
The elliptic nome is the initial point of the construction of the Lambert series.
In the theta function by Carl Gustav Jacobi the nome as an abscissa is assigned to algebraic combinations of the Arithmetic geometric mean and also the complete elliptic integral of the first kind.
Many infinite series[8] can be described easily in terms of the elliptic nome: The quadrangle represents the square number of index n, because in this way of notation the two in the exponent of the exponent would appear to small.
The nome function can be used for the definition of the complete elliptic integrals of first and second kind: In this case the dash in the exponent position stands for the derivative of the so-called theta zero value function: The elliptic functions Zeta Amplitudinis and Delta Amplitudinis can be defined with the elliptic nome function[9] easily: Using the fourth root of the quotient of the nome divided by the square function as it was mentioned above, following product series definitions[10] can be set up for the Amplitude Sine, the Counter Amplitude Sine and the Amplitude Cosine in this way: These five formulas are valid for all values k from −1 until +1.
The elliptic nome is defined as an exponential function from the negative circle number times the real period ratio.
This parameterized formula for the cube of the elliptic noun is valid for all values −1 < u < 1.
with the main alignment on the mother modulus, because this formula contains a long formulation.
The elliptic nome is defined as an exponential function from the negative circle number times the real period ratio.
On the basis of the now absolved proof a direct formula for the nome cube theorem in relation to the modulus
and in combination with the Jacobi amplitude sine shall be generated: The works Analytic Solutions to Algebraic Equations by Johansson and Evaluation of Fifth Degree Elliptic Singular Moduli by Bagis showed in their quotated works that the Jacobi amplitude sine of the third part of the complete first kind integral K solves following quartic equation: Now the parametrization mentioned above is inserted into this equation: This is the real solution of the pattern
This are the general exponentiation theorems: That theorem is valid for all natural numbers n. Important computation clues: The following Jacobi amplitude sine expressions solve the subsequent equations:
Second example: The correctness of this formula can be proved by differentiating both sides of the equation balance.
Third example: The correctness of this formula can be proved by differentiating both sides of the equation balance.