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Function Repository Resource:

 BulirschEL3

Source Notebook

Evaluate Bulirsch's incomplete elliptic integral of the third kind

Contributed by: Jan Mangaldan

ResourceFunction["BulirschEL3"][x,m,p]

gives Bulirsch's incomplete elliptic integral of the third kind .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
Bulirsch’s incomplete elliptic integral of the third kind is defined as .
When x=∞, ResourceFunction["BulirschEL3"] is referred to as a complete integral.
Argument conventions for elliptic integrals are discussed in "Elliptic Integrals and Elliptic Functions".
For certain special arguments, ResourceFunction["BulirschEL3"] automatically evaluates to exact values.
ResourceFunction["BulirschEL3"] can be evaluated to arbitrary precision.
ResourceFunction["BulirschEL3"] automatically threads over lists.

Examples

Basic Examples (1) 

Evaluate numerically:

In[1]:=
ResourceFunction["BulirschEL3"][3.4, 0.7, -1.8]
Out[1]=
In[2]:=
ResourceFunction["BulirschEL3"][\[Infinity], 0.7, -1.8]
Out[2]=

Scope (5) 

Evaluate numerically for complex arguments:

In[3]:=
ResourceFunction["BulirschEL3"][1 - 2. I, 2.5 + I, -1.8]
Out[3]=

Evaluate to high precision:

In[4]:=
ResourceFunction["BulirschEL3"][-2.`30, 1/2, 3]
Out[4]=

The precision of the output tracks the precision of the input:

In[5]:=
ResourceFunction["BulirschEL3"][-2.000000000000000000000, 1/2, 3]
Out[5]=

Simple exact results are generated automatically:

In[6]:=
ResourceFunction["BulirschEL3"][0, m, p]
Out[6]=
In[7]:=
ResourceFunction["BulirschEL3"][x, {0, 1}, 1/2]
Out[7]=

BulirschEL3 threads elementwise over lists:

In[8]:=
ResourceFunction["BulirschEL3"][{0.2, 0.3, 0.7}, 2.4, 1/3]
Out[8]=

Series expansion of BulirschEL3 at the origin:

In[9]:=
Series[ResourceFunction["BulirschEL3"][z, m, p], {z, 0, 5}]
Out[9]=

Applications (2) 

Total arc length of a cornoid:

In[10]:=
N[4/Sqrt[3] (1/ 3 ResourceFunction["BulirschEL3"][\[Infinity], 1/3, 1/4] + ResourceFunction["BulirschEL3"][\[Infinity], 1/3, 1/3] - 1/3 ResourceFunction["BulirschEL3"][\[Infinity], 1/3, 1]), 25]
Out[10]=

Compare with the result of ArcLength:

In[11]:=
ArcLength[{Cos[t] (1 - 2 Sin[t]^2), Sin[t] (1 + 2 Cos[t]^2)}, {t, 0, 2 \[Pi]}, WorkingPrecision -> 25]
Out[11]=

Visualize the solid angle subtended by a cylinder:

In[12]:=
With[{pc = {0, 2, 3/4}, r = 1, h = 3/2}, Graphics3D[{Sphere[{0, 0, 0}, r/20], Cylinder[{pc, pc + {0, 0, h}}, r]}]]
Out[12]=

Evaluate the solid angle:

In[13]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/52510be7-5435-42db-946a-21e2e5b98b36"]
Out[13]=

Compare with the result of NIntegrate:

In[14]:=
With[{pc = {0, 2, 3/4}, r = 1, h = 3/2, prec = 20}, With[{b = r^2/Norm[Most[pc]], c = r Sqrt[1 - r^2/Norm[Most[pc]]^2], d = Norm[Most[pc]], zc = Last[pc]}, NIntegrate[( 2 (h + zc) Sqrt[r^2 - x1^2])/(((d + x1)^2 + (h + zc)^2) Sqrt[ d^2 + r^2 + 2 d x1 + (h + zc)^2]), {x1, -r, -b}, WorkingPrecision -> prec] + NIntegrate[( 2 zc Sqrt[r^2 - x1^2])/(((d + x1)^2 + zc^2) Sqrt[ d^2 + r^2 + 2 d x1 + zc^2]), {x1, -b, r}, WorkingPrecision -> prec] + NIntegrate[( 2 c (d - b))/(((d - b)^2 + (z1 + zc)^2) Sqrt[ c^2 + (d - b)^2 + (z1 + zc)^2]), {z1, 0, h}, WorkingPrecision -> prec]]]
Out[14]=

Properties and Relations (2) 

Both incomplete and complete cases of EllipticPi can be expressed in terms of BulirschEL3:

In[15]:=
With[{n = 4/5, \[Phi] = \[Pi]/5, m = 2/3}, N[{EllipticPi[n, \[Phi], m], ResourceFunction["BulirschEL3"][Tan[\[Phi]], 1 - m, 1 - n]}]]
Out[15]=
In[16]:=
With[{n = 4/5, m = 2/3}, N[{EllipticPi[n, m], ResourceFunction["BulirschEL3"][\[Infinity], 1 - m, 1 - n]}]]
Out[16]=

Express EllipticF and EllipticE in terms of BulirschEL3:

In[17]:=
With[{\[Phi] = \[Pi]/5, m = 2/3}, N[{EllipticF[\[Phi], m], ResourceFunction["BulirschEL3"][Tan[\[Phi]], 1 - m, 1]}]]
Out[17]=
In[18]:=
With[{\[Phi] = \[Pi]/5, m = 2/3}, N[{EllipticE[\[Phi], m], (1 - m) ResourceFunction["BulirschEL3"][Tan[\[Phi]], 1 - m, 1 - m] + (m Sin[2 \[Phi]])/(2 Sqrt[1 - m Sin[\[Phi]]^2])}]]
Out[18]=

Requirements

Wolfram Language 12.3 (May 2021) or above

Version History

  • 1.0.0 – 29 June 2021

Source Metadata

  • Citation:
    • Bulirsch R., "Numerical calculation of elliptic integrals and elliptic functions." Numerische Mathematik, vol. 7, no. 1, 78-90, 1965. DOI: 10.1007/BF01397975
    • Bulirsch R., "An extension of the Bartky-transformation to incomplete elliptic integrals of the third kind." Numerische Mathematik, vol. 13, no. 3, 266-284, 1969. DOI: 10.1007/BF02167558
    • Bulirsch R., "Numerical calculation of elliptic integrals and elliptic functions. III." Numerische Mathematik, vol. 13, no. 4, 305-315, 1969. DOI: 10.1007/BF02165405

Related Resources

Author Notes

Requires 12.3 or later.
The form of the second argument was changed from the original definition used by Bulirsch, so that BulirschEL3 uses the parameter m=k2 instead of the modulus k. This conforms with Mathematica's choice to use the parameter in the built-in elliptic integrals and functions.

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