Function Repository Resource:

HolderDoubleSine

Source Notebook

Calculate the numerical value of the Hölder double sine function

Contributed by: Pieter-Jan De Smet

ResourceFunction["HolderDoubleSine"][x,b]

gives the Hölder double sine function s_b(x).

Details

The (Hölder) double sine function s_b(x) is defined as

, with Q=b+b^-1. Here, the double Gamma function is defined as

where the double zeta function is defined as an analytical continuation of the sum

The variable b should be real and strictly positive. The variable x can be complex.

Examples

Basic Examples (1)

Evaluate numerically:

In[1]:=

b = 1.3;
x = 1 + 2 I;
ResourceFunction["HolderDoubleSine"][x, b]

Out[3]=

Properties and Relations (5)

The double sine function is invariant under inversion of the variable b, s_b(x)=s_b^-1(x):

In[4]:=

b = RandomReal[{.1, 10}];
x = RandomComplex[{-2 - 2 I, 2 + 2 I}];
{ResourceFunction["HolderDoubleSine"][x, b], ResourceFunction["HolderDoubleSine"][x, 1/b]}

Out[6]=

The double sine function at the values ±x satisfies the relation s_b(x)s_b(-x)=1:

In[7]:=

b = RandomReal[{.1, 7}];
x = RandomComplex[{-2 - 2 I, 2 + 2 I}];
ResourceFunction["HolderDoubleSine"][x, b] ResourceFunction[
"HolderDoubleSine"][-x, b]

Out[9]=

The double sine function at z and its conjugate satisfies the relation :

In[10]:=

b = RandomReal[{.1, 8}];
z = RandomComplex[{-5 - 5 I, 5 + 5 I}];
{Conjugate[ResourceFunction["HolderDoubleSine"][z, b]], ResourceFunction["HolderDoubleSine"][-Conjugate[z], b]}

Out[12]=

The double sine function satisfies the following important difference equation :

In[13]:=

b = RandomReal[{.1, 3}];
z = RandomComplex[{-2 - 2 I, 2 + 2 I}];
{ResourceFunction["HolderDoubleSine"][z + I /2 b, b]/
ResourceFunction["HolderDoubleSine"][z - I /2 b, b], 1 / (2 Cosh[Pi b z])}

Out[15]=

Similarly, the double sine function satisfies the difference equation :

In[16]:=

Out[18]=

Possible Issues (2)

The variable b should be real and strictly positive:

In[19]:=

x = 2.2;
b = 1 + I;
ResourceFunction["HolderDoubleSine"][x, b]

Out[20]=

A warning is generated if there is a potential numerical issue:

In[21]:=

b = 1.2;
x = 50 + 1.5 I;
ResourceFunction["HolderDoubleSine"][x, b]

Out[22]=

Applications (1)

If b² is an odd integer, then the function has a closed form expression[KNB]:

In[23]:=

$Clear[x, b, CurlyD, CurlyDClosedForm] CurlyD[x_, b_] := ResourceFunction["HolderDoubleSine"][x + I / 4 ( b + 1/b), b]/ ResourceFunction["HolderDoubleSine"][x - I / 4 ( b + 1/b), b] CurlyDClosedForm[x_, n_] := With[{b = Sqrt[2 n - 1]}, Product[1 / (2 Cosh[Pi/b x + Pi I/b^2 ( ( n + 1)/2 - l)]), {l, 1, n}]]$

In[24]:=

n = 1;
b = Sqrt[2 n - 1];
x = RandomComplex[{-2 - 2 I, 2 + 2 I}];
value1 = CurlyD[x, b];
value2 = CurlyDClosedForm[x, n];
{value1, value2}

Out[29]=

In[30]:=

n = 2;
b = Sqrt[2 n - 1];
x = RandomComplex[{-2 - 2 I, 2 + 2 I}];
value1 = CurlyD[x, b];
value2 = CurlyDClosedForm[x, n];
{value1, value2}

Out[35]=

In[36]:=

n = 3;
b = Sqrt[2 n - 1];
x = RandomComplex[{-2 - 2 I, 2 + 2 I}];
value1 = CurlyD[x, b];
value2 = CurlyDClosedForm[x, n];
{value1, value2}

Out[41]=

Neat Examples (1)

has a closed form expression, see equation (A.19) in[H]:

In[42]:=

$Clear[x, b, CurlyD] CurlyD[x_, b_] := ResourceFunction["HolderDoubleSine"][x + I / 4 ( b + 1/b), b]/ ResourceFunction["HolderDoubleSine"][x - I / 4 ( b + 1/b), b] {CurlyD[0, Sqrt[2]], N[1 / ( 8 + 6 Sqrt[2])^(1/4) Exp[-Catalan/(2 Pi)]]}$

Out[44]=

Publisher

Pieter DeSmet

Requirements

Wolfram Language 13.0 (December 2021) or above

Version History

1.0.0 – 02 January 2025

Source Metadata

Citation:
- [KNP] Kubo, Nosaka, & Pang. "Exact large N expansion of mass deformed ABJM theory on squashed sphere" arXiv:2411.07334.
- [KK] Kurokawa, Koyama. "Multiple sine functions" Forum Mathematicum, vol. 15, no. 6, 2003, pp. 839-876.
- [H] Hatsuda. "ABJM on ellipsoid and topological strings" Journal of High Energy Physics 2016, no. 7 (2016): 1-30, arXiv:1601.02728.

Related Resources

Author Notes

In the literature, for example in[KK], the name double sine function is also used for a more general function.

The numerical calculation uses contour integration along a Hankel contour. The implemented contour usually works well numerically.

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License

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HolderDoubleSine

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