# Can we change Euler's Formula?

by Ahmidahn
Tags: euler, formula
 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?
 Homework Sci Advisor HW Helper Thanks ∞ PF Gold P: 11,095 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.
 P: 8 Maybe something like: (cos $\theta$ + i sin$\theta$)/w would translate into: [exp(i$\theta$)] / w where w is some decreasing function related to $\theta$? I don't know. I've been stuck on this one for almost two years. Thanks for your help.
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.
 P: 77 what if we take ei$\Theta$ and turn it into e-a+i$\Theta$? Now we have a logrithmic spiral no?.
 P: 8 Yes. That would seem to make sense. Correct me if I'm wrong reinterpreting the equation. e-a + i$\theta$ = (e-a)(ei$\theta$)= (e-a)(cos$\theta$ + i sin$\theta$) 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.
 P: 8 I'm gonna have to suss that out for the arc length...hmm...getting closer.
Homework
HW Helper
Thanks ∞
PF Gold
P: 11,095
 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, $e^{i\theta(t)}$ is rotating. You can make $a$ 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:
$$Ae^{\alpha t}e^{i\beta t} = Ae^{(\alpha+i\beta)t} = Ae^{zt}$$
if $\alpha=\beta=1$, what is the shape mapped out in the complex plane?

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

Have fun.
 P: 23 Replace $$i$$ in Euler's formula with $$i -1$$ and you get a logarithmic spiral: $$e^{(i - 1)t}$$ parametrically describes a logarithmic spiral in the complex plane
 Homework Sci Advisor HW Helper Thanks ∞ PF Gold P: 11,095 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 :)
P: 3,015
 Quote by 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?