Find a General Function for a Periodic Curve with Symmetry and Infinite Decay"

[eqm]

Hi! This is my first ever post on PF. Thanks in advance for anyone who helps me out on this!

I'm trying to find a general function that describes the curve in the attached image. As you can see, it is periodic and decays as it approaches infinity. y ≥ 0 at all times; f(0)=0; symmetric about y axis.

The period should increase as well - the distance between the first local maxima on the each side of the y-axis should be very small (nearly 0) but increases at a very large rate. For example, set the highest local maximas as p and p' and the second highest as q and q'. Well the distance between p and p' <<<< than the distance between p and q (similarly p' and q').

I'll post more drawings if need be. Thank you very much to whoever helps me out with this.

 

[eqm]

Didn't realize the picture would be huge... Sorry!

 

[Mark44]

Can you post a smaller version of the image you posted? It should be around 900 px X 600 px or so. Yours was way larger than that.

 

[eqm]

That's what I thought, sorry :/

 

[CellCoree]

Hello! Thank you for sharing your question with us. It's great to see that you are exploring mathematical functions and their properties.

Based on the description and image provided, a possible general function for this periodic curve could be:

f(x) = ae^(-bx) * sin(cx)

where a, b, and c are constants that can be adjusted to fit the specific curve in question. This function has the following properties:

1. Periodicity: The function is periodic with a period of 2π/c. As c increases, the period also increases, which aligns with the description provided.

2. Decay: The function decays as x approaches infinity, as seen in the image. The exponential term, ae^(-bx), ensures that the function approaches 0 as x approaches infinity.

3. Symmetry: The function is symmetric about the y-axis, as indicated in the description. This is because the sine function is an odd function, meaning that sin(-x) = -sin(x), which results in symmetry about the origin.

I hope this helps! Keep exploring and experimenting with different functions to gain a deeper understanding of their properties. Best of luck!