Function Repository Resource:

# UnwindingNumber

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.

## Examples

### Basic Examples (2)

Evaluate numerically:

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Plot of the unwinding number in the complex plane:

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### Scope (4)

Evaluate the unwinding number of a Root object:

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Evaluate the unwinding number of a machine precision number:

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Evaluate the unwinding number of an arbitrary precision number:

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### Applications (3)

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

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

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

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

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A relationship between the inverse sine and the inverse tangent:

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### Properties and Relations (2)

The unwinding number is an integer:

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Compare UnwindingNumber with one of its definitions:

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### Possible Issues (2)

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

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Use Simplify to resolve:

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### Neat Examples (2)

Define the Wright omega function:

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

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## Version History

• 1.0.0 – 15 March 2021