Function Repository Resource:

MakeEllipticFunction

Build an elliptic function with given periods, zeros and poles

Contributed by: Jan Mangaldan

ResourceFunction["MakeEllipticFunction"][z,{p₁,p₂},{r₁,…},{s₁,…}]

builds an elliptic function of complex argument z with periods p₁ and p₂, zeros r_i and poles s_i.

Details

The periods {p₁,p₂} define the fundamental period parallelogram for the resulting elliptic function. Their ratio must have nonzero imaginary part.

The number of zeros r_i must be equal to the number of poles s_i, and must satisfy the congruence relation

for some integers m and n.

A zero or pole of multiplicity k must be specified k times.

ResourceFunction["MakeEllipticFunction"][z,{p₁,p₂},{r₁,…},{s₁,…},f] gives an elliptic function built from the function f. Possible values of f include EllipticTheta and WeierstrassSigma, with EllipticTheta being the default.

Examples

Basic Examples (3)

Build an elliptic function over a square lattice, with two zeros and two poles:

In[1]:=

$f[z_] = ResourceFunction["MakeEllipticFunction"][ z, {2, 2 I}, {1/2, (1 + I)/2}, {0, I/2}]$

Out[1]=

Plot the real and imaginary parts of the elliptic function over a subset of the reals:

In[2]:=

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Plot the real and imaginary parts of the elliptic function over a subset of the complex plane:

In[3]:=

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Scope (2)

Form an elliptic function with a single and a double zero and a triple pole:

In[4]:=

$f[z_] = ResourceFunction["MakeEllipticFunction"][ z, {2, 2 I}, {1/3 + 1/3 I, 1/3 + 1/3 I, -2/3 - 2/3 I}, {0, 0, 0}]$

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Plot over the complex plane:

In[5]:=

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Form an elliptic function over the rhomboidal lattice with two triple zeros and one triple pole using the elliptic theta function:

In[6]:=

$f1[z_] = ResourceFunction["MakeEllipticFunction"][ z, {2, 1 + I Sqrt[3]}, {0, 0, 0, 2 + (2 I)/Sqrt[3], 2 + (2 I)/Sqrt[3], 2 + (2 I)/Sqrt[3]}, {1, (1 + I Sqrt[3])/2, (3 + I Sqrt[3])/2, 1 + I/Sqrt[3], 1 + I/Sqrt[3], 1 + I/Sqrt[3]}, EllipticTheta] // Quiet$

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Use the Weierstrass sigma function to build the elliptic function:

In[7]:=

$f2[z_] = ResourceFunction["MakeEllipticFunction"][ z, {2, 1 + I Sqrt[3]}, {0, 0, 0, 2 + (2 I)/Sqrt[3], 2 + (2 I)/Sqrt[3], 2 + (2 I)/Sqrt[3]}, {1, (1 + I Sqrt[3])/2, (3 + I Sqrt[3])/2, 1 + I/Sqrt[3], 1 + I/Sqrt[3], 1 + I/Sqrt[3]}, WeierstrassSigma] // Quiet$

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Show both functions in the complex plane:

In[8]:=

${ComplexContourPlot[{Re[f1[z]] == 0, Im[f1[z]] == 0}, {z, -3 - 3 I, 3 + 3 I}, PlotLabel -> "\[CurlyTheta]"], ComplexContourPlot[{Re[f2[z]] == 0, Im[f2[z]] == 0}, {z, -3 - 3 I, 3 + 3 I}, PlotLabel -> "\[Sigma]"]}$