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Function Repository Resource:

 BulirschEL1

Source Notebook

Evaluate Bulirsch's incomplete elliptic integral of the first kind

Contributed by: Jan Mangaldan

ResourceFunction["BulirschEL1"][x,m]

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

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
Bulirsch’s incomplete elliptic integral of the first kind is defined as .
When x=∞, ResourceFunction["BulirschEL1"] 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["BulirschEL1"] automatically evaluates to exact values.
ResourceFunction["BulirschEL1"] can be evaluated to arbitrary precision.
ResourceFunction["BulirschEL1"] automatically threads over lists.

Examples

Basic Examples (1) 

Evaluate numerically:

In[1]:=
ResourceFunction["BulirschEL1"][3.4, 0.7]
Out[1]=
In[2]:=
ResourceFunction["BulirschEL1"][\[Infinity], 0.7]
Out[2]=

Scope (5) 

Evaluate numerically for complex arguments:

In[3]:=
ResourceFunction["BulirschEL1"][1 - 2. I, 2.5 + I]
Out[3]=

Evaluate to high precision:

In[4]:=
ResourceFunction["BulirschEL1"][-2.`30, 1/2]
Out[4]=

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

In[5]:=
ResourceFunction["BulirschEL1"][-2.000000000000000000000, 1/2]
Out[5]=

Simple exact results are generated automatically:

In[6]:=
ResourceFunction["BulirschEL1"][0, m]
Out[6]=
In[7]:=
ResourceFunction["BulirschEL1"][x, {0, 1}]
Out[7]=

BulirschEL1 threads elementwise over lists:

In[8]:=
ResourceFunction["BulirschEL1"][{0.2, 0.3, 0.7}, 2.4]
Out[8]=

Series expansion of BulirschEL1 at the origin:

In[9]:=
Series[ResourceFunction["BulirschEL1"][z, m], {z, 0, 5}]
Out[9]=

Applications (2) 

Arc length of a lemniscate of Bernoulli:

In[10]:=
With[{u = 3 \[Pi]/4}, N[ResourceFunction["BulirschEL1"][Tan[u], 2] + 2 ResourceFunction["BulirschEL1"][\[Infinity], 2] Floor[ u/\[Pi] + 1/2], 25]]
Out[10]=

Compare with the result of ArcLength:

In[11]:=
With[{u = 3 \[Pi]/4}, ArcLength[{Cos[t], Sin[t] Cos[t]}/(1 + Sin[t]^2), {t, 0, u}, WorkingPrecision -> 25]]
Out[11]=

Calculate the reaction time needed for an autocatalytic termolecular reaction system:

In[12]:=
(* 80% conversion of limiting reagent *) \[Xi] = Quantity[0.8^2, ("Moles")/("Liters")]; (* equilibrium constants *) \[ScriptCapitalK]\[ScriptOne] = Quantity[2.*^-10, ("Moles")/( "Liters")]; \[ScriptCapitalK]\[ScriptTwo] = Quantity[4.*^-12, ("Moles")/( "Liters")]; \[ScriptCapitalK]\[ScriptThree] = Quantity[1.*^-8, ("Moles")/("Liters")]; (* reaction rate constant *) \[ScriptK]1 = Quantity[5.*^10, ("Liters")^2/(("Moles")^2 "Seconds")]; (* initial concentrations of A, B, C in A + B \[Rule] C + D *) A0 = Quantity[0.8, ("Moles")/("Liters")]; B0 = Quantity[1.2, ("Moles")/("Liters")]; C0 = Quantity[0.01, ("Moles")/("Liters")]; UnitConvert[ 2/(\[ScriptK]1 Sqrt[\[ScriptCapitalK]\[ScriptOne] \[ScriptCapitalK]\[ScriptTwo] \[ScriptCapitalK]\[ScriptThree]] Sqrt[ B0 + C0]) (ResourceFunction["BulirschEL1"][Sqrt[(C0 + \[Xi])/( A0 - \[Xi])], (B0 - A0)/(B0 + C0)] - ResourceFunction["BulirschEL1"][Sqrt[C0/A0], (B0 - A0)/( B0 + C0)]), "Hours"]
Out[12]=

Properties and Relations (3) 

EllipticF can be expressed in terms of BulirschEL1:

In[13]:=
With[{\[Phi] = \[Pi]/5, m = 2/3}, N[{EllipticF[\[Phi], m], ResourceFunction["BulirschEL1"][Tan[\[Phi]], 1 - m]}]]
Out[13]=

EllipticK can be expressed in terms of BulirschEL1:

In[14]:=
With[{m = 2/3}, N[{EllipticK[m], ResourceFunction["BulirschEL1"][\[Infinity], 1 - m]}]]
Out[14]=

InverseJacobiSN can be expressed in terms of BulirschEL1:

In[15]:=
With[{x = 5/8, m = 2/3}, N[{InverseJacobiSN[x, m], ResourceFunction["BulirschEL1"][x/Sqrt[1 - x^2], 1 - m]}]]
Out[15]=

Requirements

Wolfram Language 12.3 (May 2021) or above

Version History

  • 1.0.0 – 22 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

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 BulirschEL1 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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