Function Repository Resource:

ComplexRange

Source Notebook

Generate a range of complex numbers

Contributed by: Daniele Gregori

ResourceFunction["ComplexRange"][z₁]

generates a rectangular range between 0 and the complex number z₁.

ResourceFunction["ComplexRange"][{z₁}]

generates a linear range between 0 and the complex number z₁.

ResourceFunction["ComplexRange"][z₁,z₂]

generates a rectangular range between the complex numbers z₁ and z₂.

ResourceFunction["ComplexRange"][{z₁,z₂}]

generates a linear range between the complex numbers z₁ and z₂.

ResourceFunction["ComplexRange"][z₁,z₂,z₃]

generates a rectangular range between the complex numbers z₁ and z₂, with real and imaginary steps Re[z₃] and Im[z₃] respectively.

ResourceFunction["ComplexRange"][z₁,z₂,{Δz_re,Δz_im}]

generates a rectangular range between the complex numbers z₁ and z₂, with real and imaginary steps Δz_re and Δz_im respectively.

ResourceFunction["ComplexRange"][{z₁,z₂},z₃]

generates a linear range between the complex numbers z₁ and z₂, with real and imaginary steps Re[z₃] and Im[z₃] respectively.

ResourceFunction["ComplexRange"][{z₁,z₂},{Δz_re,Δz_im}]

generates a linear range between the complex numbers z₁ and z₂, with real and imaginary steps Δz_re and Δz_im respectively.

Details and Options

ResourceFunction["ComplexRange"] generalizes Range to complex numbers.

ResourceFunction["ComplexRange"] requires as first and second argument two complex numbers, spanning a range between their minimum and maximum real and imaginary parts.

A step size may optionally be given as either a complex number or a list of real numbers. By default, the imaginary part is incremented first, throughout its range, with the real part fixed; then the real part is incremented and the process is iterated until the end of the real part's range.

ResourceFunction["ComplexRange"] accepts the following options:

"IncrementFirst"

increment first the real or imaginary part

"FareyRange"

False

create a complex generalization of the Farey range

Examples

Basic Examples (2)

Create a linear range between two complex numbers:

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Visualize it in the complex plane:

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Create a rectangular range between two complex numbers:

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Visualize it in the complex plane:

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Scope (5)

Create a rectangular range between 0 and a given complex number:

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Visualize it in the complex plane:

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The range can be in any quadrant of the complex plane:

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The step subdivisions can be specified as a complex number:

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Alternatively, the third argument can be a list of the real and imaginary steps:

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It is also possible to create a linear range, by wrapping the first two arguments in a list:

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Options (6)

IncrementFirst (3)

By default, the imaginary part is incremented first:

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To increment first the real part, use the option "IncrementFirst" → Re:

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Of course, their Union is the same:

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FareyRange (3)

It is possible to specify a special range, namely the complex generalization of the resource function FareyRange:

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The third argument is then the third argument of the resource function FareyRange, possibly different for the real and imaginary parts:

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If the third argument is not a list of integers, a failure is returned:

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Properties and Relations (3)

ComplexRange[z₁,z₂,z₃] can be realized by combining Range and Outer. That is, ComplexRange[z₁,z₂,z₃] is equivalent to Flatten@Outer[#1+ⅈ#2&,Range[Re[z₁],Re[z₂],Re[z₃]],Range[Im[z₁],Im[z₂],Im[z₃]]]:

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ComplexRange[z₁,z₂,z₃] can be also realized by combining Range and Tuples. That is, ComplexRange[z₁,z₂,z₃] is equivalent to Plus@@@Tuples[{Range[Re[z₁],Re[z₂],Re[z₃], ⅈRange[Im[z₁],Im[z₂],Im[z₃]]}]:

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ComplexRange[z₁,z₂,z₃] can be also realized through CoordinateBoundsArray. That is, ComplexRange[z₁,z₂,z₃] is equivalent to Plus@@@MapAt[# ⅈ&,Flatten[CoordinateBoundsArray[{{Re[z₁],Re[z₂]},{Im[z₁],Im[z₂]}},{Re[z₃],Im[z₃]}],1],{All,2}]: