Can we change Euler's Formula?


by Ahmidahn
Tags: euler, formula
Ahmidahn
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#1
Dec19-11, 02:45 PM
P: 8
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:

y=tsint
x=tcost
z=e^(-t/12)

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?
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Simon Bridge
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#2
Dec20-11, 06:15 AM
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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.
Ahmidahn
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#3
Dec20-11, 11:41 AM
P: 8
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.

Ahmidahn
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#4
Dec20-11, 11:50 AM
P: 8

Can we change Euler's Formula?


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.
robert2734
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#5
Dec20-11, 02:53 PM
P: 77
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?.
Ahmidahn
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#6
Dec20-11, 08:05 PM
P: 8
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.
Ahmidahn
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#7
Dec20-11, 08:08 PM
P: 8
I'm gonna have to suss that out for the arc length...hmm...getting closer.
Simon Bridge
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#8
Dec20-11, 09:35 PM
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What kind of variable is "a" in this situation?
a is an angle. Anything inside the exponential function must be dimentionless.

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

Consider:
[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.
tommyli
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#9
Dec21-11, 04:04 PM
P: 23
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
Simon Bridge
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#10
Dec21-11, 06:35 PM
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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 :)
Dickfore
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#11
Dec21-11, 06:48 PM
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Quote Quote by Ahmidahn View Post
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?
Please define a "logarithmic spiral".
Ahmidahn
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#12
Dec21-11, 10:50 PM
P: 8
"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)\,
Ahmidahn
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#13
Dec21-11, 10:53 PM
P: 8
Are you wondering why I write the power in the exponent (theta) over "n"? Or what?
Ahmidahn
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#14
Dec21-11, 10:54 PM
P: 8
Thanks again for all the replies. I will be pondering this one for a while.


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