Sometimes, I get social or political ideas from mathematics. Maybe that is not such a good idea, but it is part of the meaning of this page. As the fall of the Berlin wall showed, we can have radical change in a way that is not only peaceful but also occurring in a very short time. I get that idea from this page because it shows functions that change from minus one to one in a way that is both very fast and very smooth. The other purpose of this page is that I am interested in applying Fourier series to my space physics work. The initializations are at the bottom of the page.
With ten terms, the function is as follows:
And the plot from minus ten to ten is:
We have the derivatives of all orders, and the first, with one example calculation, is:
With one hundred terms, the function is:
The first plot is from minus ten to ten, and it is starting to look like a step function.
The second plot is from minus one to one - in order to show the oscillations near zero on the horizontal axis.
The next plot shows all the oscillations along the horizontal line through the point on the vertical axis that is one unit above the horizontal axis. There are ninety-nine extrema (peaks and valleys). And ...
... the extrema above are where the first derivative is zero ... and ...
... the evident even spacing of the extrema seems confirmed by the next calculation:
With one thousand terms, the function looks more like a step function, but the oscillations are still present.
Now it seems that there are nine hundred and ninety-nine evenly spaced extrema.
As the number of terms is increased without bound, the function converges to the step function.
The function with 100 terms is numerically differentiated on the page here.