- #1

- 30

- 0

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Mappe
- Start date

- #1

- 30

- 0

- #2

- 3,332

- 2,498

A circle is symmetric and repeats.

- #3

- 212

- 49

For example, e^-x is also periodic, but its derivatives are never zero.

##e^{-x}## is not periodic (unless x is imaginary, in which case you have a real periodic function and an imaginary periodic function).

As Dr. Courtney said, check out angles of inscribed triangles within the unit circle. You keep going around and around. ##1## revolution for every ##2\pi## radians.

- #4

- 30

- 0

- #5

WWGD

Science Advisor

Gold Member

- 5,652

- 5,319

^{-i2π}= 1, and from the definition of e^{ix}by its taylor expansion, how can we see on its derivatives that its gonna be periodic and perhaps also how do we see from this definition that it describes a circle? The same question phrased differently can be "is there a simple proof that doesn't use knowledge about sin and cos, that shows us that complex numbers add their angles when multiplied, and multiplies their norm"?

If you accept the result ##e^{i\theta}= cos(\theta)+(isin\theta) ## , then this is automatic. And if you take the polar representations ( in a region where valid) of two complex numbers ##z_1=e^{i\theta_1}, z_2 = e^{i \theta_2 } ## , then ##z_1 z_2=e^{i(\theta_1+ \theta_2)} ## , which takes ##z_1=e^{i \theta_1}=cos\theta_1+i sin \theta_2## to ## z_1z_2= cos(\theta_1+ \theta_2)+ isin(\theta_1+ \theta_2) ## , which is a rotation by an angle of ##\theta_2 ##

But it ultimately depends on your starting point/assumptions.

- #6

- 1,772

- 126

http://usf.usfca.edu/vca//

He just illustrates it by example. Multiplying by i is rotating by 90 degrees, so you can draw a picture of it with that it mind for some example like (4+3i)(1+i), thinking of the complex numbers as vectors. Here, you would get 4(1+i), which which is 1+i scaled by a factor of 4 and then you get 3i(1+i), which is the 1+i rotated by 90 degrees and scaled by a factor of 3, and then you add those vectors together. So you you end up with a triangle similar to the triangle formed by 4+3i, with its real imaginary components being the other sides of the triangle that gets put on top of the one representing 1+i. When you stack the two triangles, you can see that the angles are added, and by similar triangles, the modulus gets multiplied.

It's a little cumbersome to describe verbally without a picture, so you really have to draw it and work it out for yourself.

Also, for the sine and cosine derivatives, you ought to think of taking the velocity vector of a particle that is moving around a unit circle. Some ideas along these lines are presented in Visual Complex Analysis, as well.

- #7

- 1,772

- 126

http://www.fabios.co.za/15-short-cut-extruded-pasta/trafilata-861-cavatappi/

A Cavatappi noodle can be interpreted as a 3-d plot (x,y,t) of particle moving around in a circle in the x-y plane, with time being the other coordinate. Viewed from the top, they'd look circular, but viewed from the side, they are sine waves (ignoring their thickness).

Share: