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Can we change Euler's Formula?

  1. Dec 19, 2011 #1
    Is there a way to take Euler's formula "e^(i∅)" -which gives a circle; and change it into a logarithmic spiral?

    Does a simple modification like " e^-(i∅/n) " make any sense mathematically?

    If it actually does, my other question would be; supposing that such a logarithmic spiral is in fact just a head on view of a space curve with parameters like:


    how does one go about finding the arc length of such a curve? I've attempted to use the general arc length formula on this one, but continually end up with something like:

    s = ∫ √e^-t/12 + t^2 + 1

    which is, apparently, impossible to solve.

    Is there a way to use the complex variable in a formula like Euler's formula to create a logarithmic spiral? And if so, how does one go about calculating the arc length of such a thing?
  2. jcsd
  3. Dec 20, 2011 #2

    Simon Bridge

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    Yeah - you change the amplitude with angle.

    remember - euler's formula is for a circle in the complex plane.
    the logarithmic spiral formula is for the real plane.
    to change it to complex - just multiply the y parameterization by the square-root of minus one.

    OR: you could just set Aexp(it) so that A=the radius of the spiral at angle t.

    note: exp(it/n) just changes the frequency of the rotation.
    Last edited: Dec 20, 2011
  4. Dec 20, 2011 #3
    Maybe something like:

    (cos [itex]\theta[/itex] + i sin[itex]\theta[/itex])/w

    would translate into:

    [exp(i[itex]\theta[/itex])] / w where w is some decreasing function related to [itex]\theta[/itex]? I don't know. I've been stuck on this one for almost two years.

    Thanks for your help.
  5. Dec 20, 2011 #4
    The Wikipedia entry for "Circular Polarization" has the "classical sinusoidal plane wave solution of the electromagnetic wave equation" - and the math behind it as well.

    It's similar, in general, to what I'm looking for. The math is beyond me. I actually know what some of those things are, but I think I would need to enroll in school again to understand usefully.

    Tough stuff. Maybe I should ask some physics guys how to transform Circular Polarization into Spiral Polarization.
  6. Dec 20, 2011 #5
    what if we take ei[itex]\Theta[/itex] and turn it into e-a+i[itex]\Theta[/itex]? Now we have a logrithmic spiral no?.
  7. Dec 20, 2011 #6
    Yes. That would seem to make sense. Correct me if I'm wrong reinterpreting the equation.

    e-a + i[itex]\theta[/itex] = (e-a)(ei[itex]\theta[/itex])=

    (e-a)(cos[itex]\theta[/itex] + i sin[itex]\theta[/itex])

    What kind of variable is "a" in this situation?

    Is this a three dimensional curve, or rather, can it be visualized in 3 dimensions?

    Thanks for your help there.
  8. Dec 20, 2011 #7
    I'm gonna have to suss that out for the arc length...hmm...getting closer.
  9. Dec 20, 2011 #8

    Simon Bridge

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    a is an angle. Anything inside the exponential function must be dimentionless.

    You can represent/visualize the curve how you like.

    Taken as a locus of points in the complex plane, it is two dimensional.
    If a is a constant, then the locus is a circle.

    But you can also make the angle, any of them, a function of time - in which case, [itex]e^{i\theta(t)}[/itex] is rotating. You can make [itex]a[/itex] a function of time, or even of a third space dimension if you want ... and plot a locus in 3D or an evolving spiral path in 3+1.

    [tex]Ae^{\alpha t}e^{i\beta t} = Ae^{(\alpha+i\beta)t} = Ae^{zt}[/tex]
    if [itex]\alpha=\beta=1[/itex], what is the shape mapped out in the complex plane?

    You could also look at: [itex]at[\cos(bt)+i\sin(bt)][/itex], where a and b are arbitrary constants.

    Have fun.
  10. Dec 21, 2011 #9
    Replace [tex]i[/tex] in Euler's formula with [tex]i -1[/tex] and you get a logarithmic spiral:
    [tex]e^{(i - 1)t}[/tex] parametrically describes a logarithmic spiral in the complex plane
  11. Dec 21, 2011 #10

    Simon Bridge

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    In other words, in OP notation, a=-t \theta = t.
    But you can have fun experimenting with lots of different spirals besides the golden one :)
  12. Dec 21, 2011 #11
    Please define a "logarithmic spiral".
  13. Dec 21, 2011 #12
    "Logarithmic spiral"= r = a(exp)-b(theta) in polar coordinates

    Parametric form:

    x(t) = r(t) \cos(t) = ae^{bt} \cos(t)\,
    y(t) = r(t) \sin(t) = ae^{bt} \sin(t)\,
  14. Dec 21, 2011 #13
    Are you wondering why I write the power in the exponent (theta) over "n"? Or what?
  15. Dec 21, 2011 #14
    Thanks again for all the replies. I will be pondering this one for a while.
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