On Elliptic and Hyperbolic Modular Functions and the Corresponding Gudermann Peeta Functions

In this article, we move back almost 200 years to Christoph Gudermann, the great expert on elliptic functions, who successfully put the twelve Jacobi functions in a didactic setting. We prove the second hyperbolic series expansions for elliptic functions again, and express generalizations of many of Gudermann's formulas in Carlson's modern notation. The transformations between squares of elliptic functions can be expressed as general Möbius transformations, and a conjecture of twelve formulas, extending a Gudermannian formula, is presented. In the second part of the paper, we consider the corresponding formulas for hyperbolic modular functions, and show that these Möbius transformations can be used to prove integral formulas for the inverses of hyperbolic modular functions, which are in fact Schwarz-Christoffel transformations. Finally, we present the simplest formulas for the Gudermann Peeta functions, variations of the Jacobi theta functions. 2010 Mathematics Subject Classification: Primary 33E05; Secondary 33D15.


Introduction
The elliptic integrals were first classified by Euler and Legendre, and then Gauss, Jacobi and Abel started to study their inverses, the elliptic functions.Starting in the 1830s, Gudermann published a series of papers in German and Latin, with the aim of presenting these functions in a didactic way, and to introduce a short notation for them.This notation, with a small modification, has survided until the present day.Jacobi, in 1829, had found quickly converging Fourier series expansions for most of the twelve elliptic functions, which have been put in q-hypergeometric form in the authors article [1].As Gudermann [2] showed, there are second series expansions for the twelve elliptic functions, starting from the imaginary period, which are not so quickly converging for all values of the variables; these expansions were also found, without proof, by Glaisher [3].Since these hyperbolic expansions are virtually unknown today, we prove them again, and also put them into q-hypergeometric form in section two.There are many series expansions for squareroots of rational functions of elliptic functions; as a bonus we also prove some of these.However, before this, we introduce the q-hypergeometric notation in this first section, this can also be found in the book [4].
In section three, we generalize many of Gudermanns formulas to the very general Carlson [5] notation, where many formulas can be put into one single equation by using a clever code, and the symmetry of these functions.This notation has been known for many years, but was only recently published; by coincidence, the author saw it when he was asked to review this article by Carlson.In particular, a formula with squareroots, stated without proof by Gudermann, is generalized to a conjecture of two formulas with squareroots, or twelve elliptic function formulas, which generalize four formulas with squareroots for trigonometric and hyperbolic functions.Gudermann was the first to point out the close relationship between trigonometric and hyperbolic functions.We also state four Möbius transformations in Carlson's notation, and generalize Gudermanns formulas for artanh.
In section four, we come to the hyperbolic modular functions, which have not yet appeared in the English literature; the function SN(u) is the inverse of an hyperbolic integral, which is formed by changing two minuses to plus in the elliptic integral of the first kind.We calculate the poles, periods, Möbius transformations for squares, and special values of the hyperbolic modular functions.Finally, we compute several addition formulas using a short notation for these functions.
In section five, we consider the Peeta functions, which are theta functions with imaginary function value.We show that the hyperbolic modular functions can be expressed as quotients of Peeta functions, and that the four Peeta functions are solutions of a certain heat equation with the variable q as parameter.
Before presenting the q-series formulas in the next section, we present the necessary definitions.An elliptic integral is given by where 0 < k < 1.
Abel and Jacobi, inspired by Gauss, discovered that inverting F (z) gave the doubly periodic elliptic function In connection with elliptic functions k always denotes the modulus.
Definition 1.Let δ > 0 be an arbitrarily small number.We will always use the following branch of the logarithm: −π + δ < Im (log q) ≤ π + δ.This defines a simply connected space in the complex plane.The power function is defined by q a ≡ e a log(q) (3) Definition 2. The q-factorials and the tilde operator are defined by a; q n ≡ n−1 m=0 (1 + q a+m ) (5) Definition 3. The q-hypergeometric series is defined by where It is assumed that the denominator contains no zero factors.
By √ z we mean the branch |z| 1 2 exp(i 1 2 argz).Everywhere we have y ≡ πu 2K .To maintain a symmetrical form, we put according to Jacobi and Glaisher q ≡ e −π K K .The following lemma will be used in the proofs.
2. Hyperbolic series expansions The following series were published for the first time by Gudermann in [2, p. 106], see also [7].
Proof.We only prove Equation ( 9), the other formulas are proved similarly.
Proof.All formulas are proved with the help of the previous theorem.We first observe that The Formulas ( 16) and ( 17) follow from Formula (22), the Formulas ( 18) and ( 19) follow from Formula (23) and finally, Formulas (20) and ( 21) follow from Formula (24).
Theorem 2.4.According to Heine, these 12 series can be written as follows.

Some New Elliptic Function Formulas in Carlsons Notation
Bille Carlson (1924Carlson ( -2013) ) [5] managed to simplify the great number of elliptic function formulas into a series of very general formulas.First put {p,q,r} ≡ {c,d,n} and use Glaisher's abbreviations for Jacobis elliptic functions.Thus q is not a q-analogue in this section.Furthermore, we put ( p,q) ≡ ps 2 − qs 2 , p,q ∈ {c,d,n} which implies that The default function values are u, k.All formulas apply for u ∈ Σ (Riemann sphere).It is well-known that All formulas in this section lie between these two limits, i.e., for all limits in k, we get known trigonometric and hyperbolic function (or trivial) formulas.We first give one of Carlsons results; all other formulas are presumably new.Theorem 3.1.Addition formulas [5, p. 248].Put ps i ≡ ps(u i , k), i = 1, 2, and similar notation for the other functions.Then Put ps i ≡ ps(u i , k), i = 1, 2, and similar notation for the other functions.
Theorem 3.2.Formulas for elliptic functions corresponding to product formulas for trigonometric functions.
Theorem 3.3.Formulas for elliptic functions corresponding to product formulas for trigonometric functions.
Special cases of Formula (62) were first given by Legendre [11, p. 4]  Theorem 3.4.Formulas for elliptic functions corresponding to product formulas for trigonometric functions.
Theorem 3.7.For p = n, Formula (68) holds unaltered.In the two other cases we only use one of the two factors qn, rn in either numerator or denominator.This gives the six formulas Special case of Formula (68) were given by Gudermann 1838 [10, (8), ( 9), (11) p. 157].
Special cases of the following formulas were given by Gudermann 1838 [10, p. 148 f].
Half of the following conjecture was given in [2, p. 109].We have the well-known formulas Conjecture 3.10.We have the twelve formulas where f, g ∈ {qs, ps}.We choose the right hand side that has the correct limits for lim k→0 + and lim k→1 − in Formulas ( 72)-(75).Ten of the formulas have one limit among these four formulas, and the remaining two (with ncu) have two limits.
To be able to compute the inverses of the elliptic functions, we must first prove a number of Möbius transformations between squares of elliptic functions which govern their transformations.Most of these can be summarized in the formulas pq 2 = (p,r)qr 2 + (q,p) (q,r)qr 2 (81) Formula (81) is its own inverse.With these formulas we can prove integral formulas for the twelve inverse elliptic functions like in [14, p. 102] and [15, 22.15].The integral formulas for the inverses of the elliptic functions are Schwarz-Christoffel mappings from the periodic rectangle of each elliptic function.The Formulas (78) to (81) do not map each periodic rectangle to the next, but if we take all rectangles in the vicinity of the origin and agree to start each equation solving with the prerequisite (z) > 0, the formulas give correct values on the Riemann sphere.
We conclude with a few formulas with the function artanh(x), which again generalize Gudermanns results.
Theorem 3.11.We have the eighteen formulas We find that Furthermore, we put Definition 7. The Gudermannian function l(x) relates the circular functions and hyperbolic functions without using complex numbers.It is given by The inverse function or the Mercator function is given by The function L(x) is the inverse of l(x).Legendre calculated tables for this function.Since Lϕ > ϕ it follows that [9, p. 27] We also find that cosh u < CNu < cosh Lu, tanh u < SCu < tanh Lu (98) The hyperbolic elliptic functions can also be transformed to the hyperbolic potential functions by putting SNu = sinhψ, CNu = cosh ψ and SCu = tanh ψ (99) The arc ψ is is called the hyperbolic amplitude of the argument u for the module kt; or ψ = Am u, and vice versa u = Arg Am (ψ).
Next we have [9, p. 27] Below is a list of the inverse of the four hyperbolic modular functions: When t = SNu, so u = ArgSNt; When t = CNu, so u = ArgCNt; When t = SCu, so u = ArgSCt; When t = DNu, so u = ArgDNt; We have DAmu = DNu (102) Formula ( 87) is equivalent to The poles and periods are shown in the following table : half period poles iK poles K + iK poles K poles 0 We have the following special values for the twelve hyperbolic modular functions: Just like the trigonometric and hyperbolic potential functions can be transformed to each other by multiplication by i, the trigonometric and hyperbolic modular functions can also be mapped to each other with utter avoidance of imaginary forms [9, p. 31].These transformations look like this (we use the Glaisher notation and means the module k ): Proof.Use Formula (105).Special cases of Formulas (128) to (131) were first given by Gudermann 1838 [9, p. 33].

The Peeta Functions
The Peeta functions were first introduced by Gudermann [18, p. 79], to be able to express the hyperbolic modular functions by theta functions in a manner similar to the elliptic functions.
Eagle [7, p. 81] has also spoken of the great importance of these functions.
The four Peeta functions are defined as follows: Definition 8.