Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 UnwindingNumber

Source Notebook

Evaluate the unwinding number

Contributed by: Jan Mangaldan

ResourceFunction["UnwindingNumber"][z]

gives the unwinding number 𝒰(z).

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
The unwinding number is defined by the relation z=log ez+2πi𝒰​(z).
ResourceFunction["UnwindingNumber"][z] returns an integer when z is any numeric quantity, whether or not it is an explicit number.
For exact numeric quantities, ResourceFunction["UnwindingNumber"] internally uses numerical approximations to establish its result. This process can be affected by the setting of the global variable $MaxExtraPrecision.
ResourceFunction["UnwindingNumber"] automatically threads over lists.

Examples

Basic Examples (2) 

Evaluate numerically:

In[1]:=
ResourceFunction["UnwindingNumber"][3 - 4 I]
Out[1]=
Image

Plot of the unwinding number in the complex plane:

In[2]:=
ComplexPlot3D[ResourceFunction["UnwindingNumber"][z], {z, 4 \[Pi]}, ColorFunction -> {ColorData[97, 2] &, None}, Mesh -> True, MeshFunctions -> {Re[#1] &, Im[#1] &}]
Out[2]=
Image

Scope (4) 

Evaluate the unwinding number of a Root object:

In[3]:=
ResourceFunction["UnwindingNumber"][ Root[-22400000 - 480000 # + #^5& , 3, 0]]
Out[3]=
Image

Evaluate the unwinding number of a machine precision number:

In[4]:=
ResourceFunction["UnwindingNumber"][N[2 ArcSin[10]]]
Out[4]=
Image

Evaluate the unwinding number of an arbitrary precision number:

In[5]:=
ResourceFunction["UnwindingNumber"][N[2 ArcSin[10], 25]]
Out[5]=
Image

UnwindingNumber threads elementwise over lists:

In[6]:=
ResourceFunction["UnwindingNumber"][ RandomComplex[{-1, 1} 4 \[Pi] (1 + I), 5]]
Out[6]=
Image

Applications (3) 

The identity Image does not generally hold for complex z and w:

In[7]:=
Sqrt[z w] == Sqrt[z] Sqrt[w] /. {z -> -1 - I, w -> 2 - 3 I} // FullSimplify
Out[7]=
Image

Use the unwinding number to construct a formula that is valid in the entire complex plane:

In[8]:=
Sqrt[z w] == Sqrt[z] Sqrt[w] (-1)^ ResourceFunction["UnwindingNumber"][Log[z] + Log[w]] /. {z -> -1 - I, w -> 2 - 3 I} // FullSimplify
Out[8]=
Image

The identity Image does not generally hold for complex z and w:

In[9]:=
Log[z w] == Log[z] + Log[w] /. {z -> -1 - I, w -> 2 - 3 I} // FullSimplify
Out[9]=
Image

Use the unwinding number to construct a formula that is valid in the entire complex plane:

In[10]:=
Log[z w] == Log[z] + Log[w] - 2 \[Pi] I ResourceFunction["UnwindingNumber"][ Log[z] + Log[w]] /. {z -> -1 - I, w -> 2 - 3 I} // FullSimplify
Out[10]=
Image

A relationship between the inverse sine and the inverse tangent:

In[11]:=
ArcSin[z] == ArcTan[z/Sqrt[ 1 - z^2]] + \[Pi] (ResourceFunction[ "UnwindingNumber"][-Log[1 + z]] - ResourceFunction["UnwindingNumber"][-Log[1 - z]]) /. z -> -5.1
Out[11]=
Image

Properties and Relations (2) 

The unwinding number is an integer:

In[12]:=
ResourceFunction[ "UnwindingNumber"][{Cosh[5 + 3 I], Root[#^11 - 19 # + 13 &, 4]}]
Out[12]=
Image

Compare UnwindingNumber with one of its definitions:

In[13]:=
ResourceFunction["UnwindingNumber"][z] == (z - Log[E^z])/( 2 \[Pi] I) /. z -> RandomComplex[5 \[Pi] (1 + I) {-1, 1}, WorkingPrecision -> $MachinePrecision] // FullSimplify
Out[13]=
Image

Possible Issues (2) 

Numerical decision procedures with default settings cannot automatically resolve this value:

In[14]:=
ResourceFunction["UnwindingNumber"][ 1 + I ((3 \[Pi] + 4 \[Pi]^2 + 2 \[Pi]^3) - 2 \[Pi] (\[Pi] + 1)^2)]
Image
Out[14]=
Image

Use Simplify to resolve:

In[15]:=
Simplify[%]
Out[15]=
Image

Neat Examples (2) 

Define the Wright omega function:

In[16]:=
wrightOmega[z_] := ProductLog[ResourceFunction["UnwindingNumber"][z], Exp[z]]

Visualize the fringing fields of a semi-infinite parallel plate capacitor:

In[17]:=
ParametricPlot[ With[{z = x + I y}, ReIm[z - 1 - wrightOmega[z - 1]]], {x, -10, 10}, {y, -2 \[Pi], 2 \[Pi]}, Sequence[ Axes -> None, BoundaryStyle -> None, Mesh -> 30, MeshFunctions -> {#4& }, MeshStyle -> Opacity[0.6, RGBColor[0.368417, 0.506779, 0.709798]], PlotPoints -> 75, PlotRange -> {{-7, 2}, {-Pi, Pi}}, PlotStyle -> None]]
Out[17]=
Image

Version History

  • 1.0.0 – 15 March 2021

Source Metadata

  • Citation:
    • Bradford R., Corless R.M., Davenport J.H., Jeffrey D.J., Watt S.M., "Reasoning about the Elementary Functions of Complex Analysis." Annals of Mathematics and Artificial Intelligence, vol. 36, 303-318, 2002. DOI: 10.1023/A:1016007415899
    • Corless R.M., Jeffrey D.J., "The Unwinding Number." SIGSAM Bulletin, vol. 30, 28-35, 1996. DOI: 10.1145/235699.235705

Author Notes

The sign convention used by UnwindingNumber is opposite from the one used in the paper of Corless and Jeffrey that originally introduced the concept of the unwinding number.

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