Can we change Euler's Formula?

Ahmidahn

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?

Simon Bridge

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

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

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

what if we take e^{i[itex]\Theta[/itex]} and turn it into e^{-a+i[itex]\Theta[/itex]}? Now we have a logrithmic spiral no?.

Ahmidahn

Yes. That would seem to make sense. Correct me if I'm wrong reinterpreting the equation.

e^{-a + i[itex]\theta[/itex]} = (e^-a)(e^{i[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

I'm gonna have to suss that out for the arc length...hmm...getting closer.

Simon Bridge

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.

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

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

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

Please define a "logarithmic spiral".

Ahmidahn

"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

Are you wondering why I write the power in the exponent (theta) over "n"? Or what?

Ahmidahn

Thanks again for all the replies. I will be pondering this one for a while.