Helix

(Facebook/Mathematics/BM)

exp(ix) is a complex-valued function of a real variable. It is periodic, with period 2π, so the graph in ℝxℂ will be a spiral (helix).

How will be xix? (Sourav Hom Choudhury)

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xix = eix·ln x = e ix·(ln|x| + i·arg x + n·2π·i)

= eix·ln|x| · e-x·arg x · e-2π·n·x =

= e-x·arg x · e-2π·n·x · (cos(x·ln|x|) + i sin(x·ln|x|))

I suppose you mean that x is real. Yet, the function is multivalued, with one branch for each integer n. arg x is 0 or π depending on whether x is positive or negative.

Example: For n = 0 we have a branch with

xix = e-x·arg x · ( cos(x·ln|x|) + i sin(x·ln|x|) )

The graph in ℝxℂ will be some kind of spiral, since the e-factor has period p, given by p·ln p = 2π, or p ≈ 4.304530324, and describes a circle in ℂ when x goes throught an interval of length p. For x > 0 this gives a spiral (helix) with constant radius 1 but for x < 0 the radius varies since e-x·arg x = eπ·x.

BTW The exact value of p is given by p = eW₀(2π), where W₀ is (one branch of) our old friend, the Lambert W-function. This function once more – amazing! 🙂

Correction: The e-factor is not periodic, since the function x → x·ln|x| is not linear! Certainly the graph of xix is a spiral in ℝxℂ but the x-interval corresponding to one revolution is not constant. Also near x = 0 the direction will be reversed since x·ln|x| changes sign. – A fascinating curve!

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The graph in RxC of one branch of the function
x -> xix. For x > 0 the width of the spiral is constant, for x < 0 it increases exponentially with decreasing x. Also the revolutions do not have constant height, but they change slowly. The proportions are not exactly correct, in order to get a better view.

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Magnification of a part near the origin. Passing from positive to negative x the curve makes a little ”step backwards”.

These two images were calculated and plotted using Mathematica.

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The spiral viewed from a point straight below 0 in the complex plane. Notice the little change of direction (twice) near 1.

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The real and imaginary parts of xix, as functions of the real variable x. The Im-curve shows how the direction of rotation with changing x reverses direction twice around the origin.

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