In mathematics, specifically the theory of elliptic functions, the nome is a special function that belongs to the non-elementary functions. This function is of great importance in the description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents and the Weber modular functions, that are used for solving equations of higher degrees.

Definition

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The nome function is given by

 

where   and   are the quarter periods, and   and   are the fundamental pair of periods, and   is the half-period ratio. The nome can be taken to be a function of any one of these quantities; conversely, any one of these quantities can be taken as functions of the nome. Each of them uniquely determines the others when  . That is, when  , the mappings between these various symbols are both 1-to-1 and onto, and so can be inverted: the quarter periods, the half-periods and the half-period ratio can be explicitly written as functions of the nome. For general   with  ,   is not a single-valued function of  . Explicit expressions for the quarter periods, in terms of the nome, are given in the linked article.

Notationally, the quarter periods   and   are usually used only in the context of the Jacobian elliptic functions, whereas the half-periods   and   are usually used only in the context of Weierstrass elliptic functions. Some authors, notably Apostol, use   and   to denote whole periods rather than half-periods.

The nome is frequently used as a value with which elliptic functions and modular forms can be described; on the other hand, it can also be thought of as function, because the quarter periods are functions of the elliptic modulus  :  .

The complementary nome   is given by

 

Sometimes the notation   is used for the square of the nome.

The mentioned functions   and   are called complete elliptic integrals of the first kind. They are defined as follows:

 
 

Applications

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The nome solves the following equation:

 

This analogon is valid for the Pythagorean complementary modulus:

 

where   are the complete Jacobi theta functions and   is the complete elliptic integral of the first kind with modulus   shown in the formula above. 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. That is, the half-period ratio   is commonly used as a coordinate on the complex upper half-plane, typically endowed with the Poincaré metric to obtain the Poincaré half-plane model. The nome then serves as a coordinate on a punctured disk of unit radius; it is punctured because   is not part of the disk (or rather,   corresponds to  ). This endows the punctured disk with the Poincaré metric.

The upper half-plane (and the Poincaré disk, and the punctured disk) can thus be tiled with the fundamental domain, which is the region of values of the half-period ratio   (or of  , or of   and   etc.) that uniquely determine a tiling of the plane by parallelograms. The tiling is referred to as the modular symmetry given by the modular group. Some functions that are periodic on the upper half-plane are called to as modular functions; the nome, the half-periods, the quarter-periods or the half-period ratio all provide different parameterizations for these periodic functions.

The prototypical modular function is Klein's j-invariant. It can be written as a function of either the half-period ratio τ or as a function of the nome  . The series expansion in terms of the nome or the square of the nome (the q-expansion) is famously connected to the Fisher-Griess monster by means of monstrous moonshine.

Euler's function arises as the prototype for q-series in general.

The nome, as the   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.

Curve sketching

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Every real value   of the interval   is assigned to a real number between inclusive zero and inclusive one in the nome function  . The elliptic nome function is axial symmetric to the ordinate axis. Thus:  . The functional curve of the nome passes through the origin of coordinates with the slope zero and curvature plus one eighth. For the real valued interval   the nome function   is strictly left-curved.

Derivatives

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The Legendre's relation is defined that way:

 

And as described above, the elliptic nome function   has this original definition:

 

Furthermore, these are the derivatives of the two complete elliptic integrals:

 
 

Therefore, the derivative of the nome function has the following expression:

 

The second derivative can be expressed this way:

 

And that is the third derivative:

 

The complete elliptic integral of the second kind is defined as follows:

 

The following equation follows from these equations by eliminating the complete elliptic integral of the second kind:

 

Thus, the following third-order quartic differential equation is valid:

 

MacLaurin series and integer sequences

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Kneser sequence

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Given is the derivative of the Elliptic Nome mentioned above:

 

The outer factor with the K-integral in the denominator shown in this equation is the derivative of the elliptic period ratio. The elliptic period ratio is the quotient of the K-integral of the Pythagorean complementary modulus divided by the K-integral of the modulus itself. 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. The Kneser integer sequence Kn(n) can be constructed in this way:

 

 

Executed examples:

 

 

 

 

 

 

The Kneser sequence appears in the Taylor series of the period ratio (half period ratio):

 
 

The derivative of this equation after   leads to this equation that shows the generating function of the Kneser number sequence:

 
 

This result appears because of the Legendre's relation   in the numerator.

Schellbach Schwarz sequence

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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. The construction of this sequence is detailed in his work Die Lehre von den Elliptischen Integralen und den Thetafunktionen.[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. The mathematician Adolf Kneser determined a construction for this sequence based on the following pattern:

 

The Schellbach Schwarz sequence Sc(n) appears in the On-Line Encyclopedia of Integer Sequences under the number A002103 and the Kneser sequence Kn(n) appears under the number A227503.

The following table[3][4] contains the Kneser numbers and the Schellbach Schwarz numbers:

Constructed sequences Kneser and Schellbach Schwarz
Index n Kn(n) (A227503) Sc(n) (A002103)
1 1 1
2 13 2
3 184 15
4 2701 150
5 40456 1707
6 613720 20910
7 9391936 268616
8 144644749 3567400

And this sequence creates the MacLaurin series of the elliptic nome[5][6][7] in exactly this way:

 
 

In the following, it will be shown as an example how the Schellbach Schwarz numbers are built up successively. For this, the examples with the numbers Sc(4) = 150, Sc(5) = 1707 and Sc(6) = 20910 are used:

 
 
 
 
 
 

Kotěšovec sequence

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The MacLaurin series of the nome function   has even exponents and positive coefficients at all positions:

 

And the sum with the same absolute values of the coefficients but with alternating signs generates this function:

 

The radius of convergence of this Maclaurin series is 1. Here   (OEIS A005797) is a sequence of exclusively natural numbers   for all natural numbers   and this integer number sequence is not elementary. This sequence of numbers   was researched by the Czech mathematician and fairy chess composer Václav Kotěšovec, born in 1956. Two ways of constructing this integer sequence shall be shown in the next section.

Construction method with Kneser numbers

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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   anymore, but   instead of that:

 

Following table contains the Schellbach Schwarz numbers and the Kneser numbers and the Apéry numbers:

Constructed sequences Kneser and Kotěšovec
Index n Kn(n) (A227503) Kt(n) (A005797)
1 1 1
2 13 8
3 184 84
4 2701 992
5 40456 12514
6 613720 164688
7 9391936 2232200
8 144644749 30920128

In the following, it will be shown as an example how the Schellbach Schwarz numbers are built up successively. For this, the examples with the numbers Kt(4) = 992, Kt(5) = 12514 and Kt(6) = 164688 are used:

 
 
 
 
 
 

So the MacLaurin series of the direct Elliptic Nome can be generated:

 
 

Construction method with Apéry numbers

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By adding a further integer number sequence   that denotes a specially modified Apéry sequence (OEIS A036917), the sequence of the Kotěšovec numbers   can be generated. The starting value of the sequence   is the value   and the following values of this sequence are generated with those two formulas that are valid for all numbers  :

 
 

This formula creates the Kotěšovec sequence too, but it only creates the sequence numbers of even indices:

 

The Apéry sequence   was researched especially by the mathematicians Sun Zhi-Hong and Reinhard Zumkeller. And that sequence generates the square of the complete elliptic integral of the first kind:

 

The first numerical values of the central binomial coefficients and the two numerical sequences described are listed in the following table:

Index n Central binomial coefficient square   Sequence number Ap(n) Sequence number Kt(n)
1 1 1 1
2 4 8 8
3 36 88 84
4 400 1088 992
5 4900 14296 12514
6 63504 195008 164688
7 853776 2728384 2232200
8 11778624 38879744 30920128
9 165636900 561787864 435506703
10 2363904400 8206324928 6215660600
11 34134779536 120929313088 89668182220
12 497634306624 1794924383744 1305109502496
13 7312459672336 26802975999424 19138260194422
14 108172480360000 402298219288064 282441672732656
15 1609341595560000 6064992788397568 4191287776164504
16 24061445010950400 91786654611673088 62496081197436736
17 361297635242552100 1393772628452578264 935823746406530603

Václav Kotěšovec wrote down the number sequence   on the Online Encyclopedia of Integer Sequences up to the seven hundredth sequence number.

Here one example of the Kotěšovec sequence is computed:

 

 

 

Function values

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The two following lists contain many function values of the nome function:

The first list shows pairs of values with mutually Pythagorean complementary modules:

 
 
 
 
 
 
 
 
 
 
 
 
 

The second list shows pairs of values with mutually tangentially complementary modules:

 
 
 
 
 
 
 
 
 
 
 
 
 

Related quartets of values are shown below:

 

 

 

 

 

 

 

 

 

 

 

 

Sums and products

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Sum series

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The elliptic nome was explored by Richard Dedekind and this function is the fundament in the theory of eta functions and their related functions. 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. So this formula is valid:  

The letter   describes the complete elliptic integral of the second kind, which is the quarter periphery of an ellipse in relation to the bigger half axis of the ellipse with the numerical eccentricity   as abscissa value.

Product series

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The two most important theta functions can be defined by following product series:

 
 

Furthermore, these two Pochhammer products have those two relations:

 
 

The Pochhammer products have an important role in the pentagonal number theorem and its derivation.

Relation to other functions

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Complete elliptic integrals

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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:

 

Definitions of Jacobi functions

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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.

Then following successive definition of the other Jacobi functions is possible:

 
 
 
 

The product definition of the amplitude sine was written down in the essay π and the AGM by the Borwein brothers on page 60 and this formula is based on the theta function definition of Whittaker und Watson.

Identities of Jacobi Amplitude functions

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In combination with the theta functions the nome gives the values of many Jacobi amplitude function values:

 
 
 
 
 
 
 

The abbreviation sc describes the quotient of the amplitude sine divided by the amplitude cosine.

Theorems and Identities

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Derivation of the nome square theorem

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The law for the square of the elliptic noun involves forming the Landen daughter modulus:

 

The Landen daughter modulus is also the tangential counterpart of the Pythagorean counterpart of the mother modulus.

This formula results as a combination of the following equations:
 

The differential quotient of this equation balance along with   confirms the correctness of this formula. Because on both sides of the equation scale the differential quotient along w is the same and the functions on both sides of the scale run through the coordinate origin with respect to w.

The next equation follows directly from the previous equation:

 

By changing the substitution this expression is generated:

 

The combination of both formulas leads to that quotient equation:

 

Both sides of this equation scale show period ratios.

For on both sides of this balance the modulus in the numerator is Pythagorean complementary to the modulus in the denominator.

The elliptic nome is defined as an exponential function from the negative circle number times the real period ratio.

And the real period ratio is defined as the quotient of the K integral of the Pythagorean complementary modulus divided by the K integral of the modulus itself.

This is the consequence:

 
 

QUOD ERAT DEMONSTRANDUM!

Examples for the nome square theorem

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The Landen daughter modulus[11][12] is identical to the tangential opposite of the Pythagorean opposite of the mother modulus.

Three examples shall be shown in the following:

Trigonometrically displayed examples:

 
 
 
 

Hyperbolically displayed examples:

 
 
 
 
 
 
 
 

Derivation of the parametrized nome cube theorem

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Not only the law for the square but also the law for the cube of the elliptic nome leads to an elementary modulus transformation. This parameterized formula for the cube of the elliptic noun is valid for all values −1 < u < 1.

 

This formula was displayed exactly like this and this time it was not printed exactly after the expression   with the main alignment on the mother modulus, because this formula contains a long formulation. And in the formula shown now with the parameter  , a greatly simplified formula emerges.

This formula results as a combination of the following equations:
 
 

The differential quotient of this equation balance along with   confirms the correctness of this formula. Because on both sides of the equation scale the differential quotient along w is the same and the functions on both sides of the scale run through the coordinate origin with respect to w.

The next equation follows directly from the previous equation:

 

By changing the substitution this expression is generated:

 

The combination of both formulas leads to that quotient equation:

 

Both sides of this equation scale show period ratios.

For on both sides of this balance the modulus in the numerator is Pythagorean complementary to the modulus in the denominator.

The elliptic nome is defined as an exponential function from the negative circle number times the real period ratio.

And the real period ratio is defined as the quotient of the K integral of the Pythagorean complementary modulus divided by the K integral of the modulus itself.

This is the consequence:

 
 
 

QUOD ERAT DEMONSTRANDUM!

Derivation of the direct nome cube theorem

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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   of that quartic equation:

 

Therefore, following formula is valid:

 

The parametrized nome cube formula has this mentioned form:

 

The same formula can be designed in this alternative way:

 

So this result appears as the direct nome cube theorem:

 

Examples for the nome cube theorem

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Alternatively, this formula can be set up:

 

The now presented formula is used for simplified computations, because the given elliptical modulus can be used to determine the value   in an easy way. The value   can be evoked by taking the tangent duplication of the modulus and then taking the cube root of that in order to get the parameterization value   directly.

Two examples are to be treated exemplarily:

In the first example, the value   is inserted:

 
 

In the second example, the value   is inserted:

 
 
 

The constant   represents the Golden ratio number   exactly. Indeed, the formula for the cube of the nome involves a modulus transformation that really contains elementary cube roots because it involves the solution of a regular quartic equation. However the laws for the fifth power and the seventh power of the elliptic nome do not lead to an elementary nome transformation, but to a non elementary transformation. This was proven by the Abel–Ruffini theorem[13][14][15] and by the Galois theory[16] too.

Exponentiation theorems with Jacobi amplitude functions

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Every power of a nome of a positive algebraic number as base and a positive rational number as exponent is equal to a nome value of a positive algebraic number:

 

These are the most important examples of the general exponentiation theorem:

 
 
 
 
 
 
 
 

The abbreviation   stands for the Jacobi elliptic function amplitude sine.

For algebraic   values in the real interval   the shown amplitude sine expressions are always algebraic.

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:

Thirds of the K:

  solves the equation[17]

 

Fifths of the K:

  solves the equation[18][19]

 

Sevenths of the K:

  solves the equation

  und

 

Elevenths of the K:

  solves the equation

  

Examples for the exponentiation theorems

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For these nome power theorems important examples shall be formulated:

Given is the fifth power theorem:

 

Lemniscatic example for the fifth power theorem:

 

 

 

 

A next example for the fifth power theorem:

 

 

 

 

 

 

Reflection theorems

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If two positive numbers   and   are Pythagorean opposites to each other and thus the equation   is valid, then this relation is valid:

 

If two positive numbers   and   are tangential opposites to each other and thus the equation   is valid, then that relation is valid:

 

Therefore, these representations have validity for all real numbers x:

Pythagorean opposites:

 
 

Tangential opposites:

 
 

Derivations of the nome values

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Direct results of mentioned theorems

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The following examples should be used to determine the nouns:

Example 1: Given is the formula of the Pythagorean counterparts:

 

For x = 0, this formula gives this equation:

 
 

Example 2:

Given is the formula of the tangential counterparts:

 

For x = 0, the formula for the tangential counterparts gives the following equation:

 
 

Combinations of two theorems each

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Example 1: Equianharmonic case

The formula of the Pythagorean counterparts is used again:

 

For  , this equation results from this formula:

 

In a previous section this theorem was stated:

 

From this theorem for cubing, the following equation results for  :

 

The solution to the system of equations with two unknowns then reads as follows:

 
 

Example 2: A further case with the cube formula

The formula of the tangential counterparts is used again:

 

For   this formula results in the following equation:

 

The theorem for cubing is also used here:

 

From the previously mentioned theorem for cubing, the following equation results for  :

 

The solution to the system of equations with two unknowns then reads as follows:

 
 

Investigations about incomplete integrals

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With the incomplete elliptic integrals of the first kind, the values of the elliptic noun function can be derived directly.

With two accurate examples, these direct derivations are to be carried out in the following:

First example:

 

The correctness of this formula can be proved by computing the differential quotient after the variable   on both sides of the balance of equation.

Using the value   gives this result:

 

The following two results emerge:

 

 

Second example:

 

The correctness of this formula can be proved by differentiating both sides of the equation balance.

 

The following two results emerge:

 

 

Third example:

 

The correctness of this formula can be proved by differentiating both sides of the equation balance.

Using the value   gives this result:

 

The following two results emerge:

 

 

First derivative of the theta function

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Derivation of the derivative

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The first derivative of the principal theta function among the Jacobi theta functions can be derived in the following way using the chain rule and the derivation formula of the elliptic nome:

 
 

For the now mentioned derivation part this identity is the fundament:

 

Therefore, this equation results:

 

The complete elliptic integrals of the second kind have that identity:

 

Along with this modular identity, following formula transformation can be made:

 

Furthermore, this identity is valid:

 

By using the theta function expressions ϑ00(x) and ϑ01(x) following representation is possible:

 

This is the final result:

 
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In a similar way following other first derivatives of theta functions and their combinations can also be derived:

 
 
 
 
 

Important definition:

 
 

References

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  1. ^ Karl Heinrich Schellbach (1864), Die Lehre von den Elliptischen Integralen und den ThetaFunctionen [The Teaching of Elliptic Integrals and Theta Functions], Berlin: G. Reimer, ISBN 978-3-11-169377-4, retrieved 2023-06-11
  2. ^ Hermann Amandus Schwarz (1885), Formeln und Lehrsätze zum gebrauche der elliptischen Functionen [Formulas and Lectures on Use of the Elliptic Function], Göttingen: W. Fr. Kaestner, ISBN 978-3-662-25776-0, retrieved 2024-04-04
  3. ^ Adolf Kneser (1927), "Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen.", Journal für die reine und angewandte Mathematik, vol. 158, pp. 209–218, ISSN 0075-4102, retrieved 2023-06-11
  4. ^ D. K. Lee (1989-03-01), Application of theta functions for numerical evaluation of complete elliptic integrals of the first and second kinds, Oak Ridge, TN (United States): Oak Ridge National Lab. (ORNL), OSTI 6137964, retrieved 2023-06-11
  5. ^ "A002103 - OEIS". Retrieved 2023-05-28.
  6. ^ "Series Expansion of EllipticNomeQ differs from older Mathematica Version". Retrieved 2023-05-28.
  7. ^ R. B. King, E. R. Canfield (1992-08-01), "Icosahedral symmetry and the quintic equation", Computers & Mathematics with Applications, vol. 24, no. 3, pp. 13–28, doi:10.1016/0898-1221(92)90210-9, ISSN 0898-1221
  8. ^ "Table of Infinite Products Infinite Sums Infinite Series Elliptic Theta". Retrieved 2021-09-30.
  9. ^ Eric W. Weisstein. "Jacobi Theta Functions". Retrieved 2021-10-01.
  10. ^ Alvaro H. Salas, Lorenzo J. H. Martinez, David L. R. Ocampo R. (2021-10-11), "Approximation of Elliptic Functions by Means of Trigonometric Functions with Applications", Mathematical Problems in Engineering, vol. 2021, pp. e5546666, doi:10.1155/2021/5546666, ISSN 1024-123X{{citation}}: CS1 maint: multiple names: authors list (link)
  11. ^ Gauss, C. F.; Nachlass (1876). "Arithmetisch geometrisches Mittel, Werke, Bd. 3". Königlichen Gesell. Wiss., Göttingen: 361–403.
  12. ^ Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
  13. ^ Ruffini, Paolo (1813). Riflessioni intorno alla soluzione delle equazioni algebraiche generali opuscolo del cav. dott. Paolo Ruffini ... (in Italian). presso la Societa Tipografica.
  14. ^ Abel, Niels Henrik (1881) [1824], "Mémoire sur les équations algébriques, ou l'on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré" (PDF), in Sylow, Ludwig; Lie, Sophus (eds.), Œuvres Complètes de Niels Henrik Abel (in French), vol. I (2nd ed.), Grøndahl & Søn, pp. 28–33
  15. ^ Abel, Niels Henrik (1881) [1826], "Démonstration de l'impossibilité de la résolution algébrique des équations générales qui passent le quatrième degré" (PDF), in Sylow, Ludwig; Lie, Sophus (eds.), Œuvres Complètes de Niels Henrik Abel (in French), vol. I (2nd ed.), Grøndahl & Søn, pp. 66–87
  16. ^ Tignol, Jean-Pierre (2001). Galois' Theory of Algebraic Equations. World Scientific. pp. 232–3, 302. ISBN 978-981-02-4541-2.
  17. ^ Viktor Prasolov and Yuri Soloviev (1997), "Elliptic Functions and Elliptic Integrals" (PDF), arXiv: General Mathematics, retrieved 2023-06-24
  18. ^ N. Bagis (2012-02-22), "Evaluation of Fifth Degree Elliptic Singular Moduli", arXiv: General Mathematics, S2CID 53372341
  19. ^ Tomas Johansson (1998-06-12), "Analytic Solutions to Algebraic Equations" (PDF), arXiv: General Mathematics, retrieved 2023-06-24
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