Introduction

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.

Terms 10

With ten terms, the function is as follows:

[Graphics:Images/index_gr_1.gif]
[Graphics:Images/index_gr_2.gif]

And the plot from minus ten to ten is:

[Graphics:Images/index_gr_3.gif]

We have the derivatives of all orders, and the first, with one example calculation, is:

[Graphics:Images/index_gr_4.gif]
[Graphics:Images/index_gr_5.gif]

Terms 100

With one hundred terms, the function is:

[Graphics:Images/index_gr_6.gif]
[Graphics:Images/index_gr_7.gif]

The first plot is from minus ten to ten, and it is starting to look like a step function.

[Graphics:Images/index_gr_8.gif]

The second plot is from minus one to one - in order to show the oscillations near zero on the horizontal axis.

[Graphics:Images/index_gr_9.gif]

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 ...

[Graphics:Images/index_gr_10.gif]

... the extrema above are where the first derivative is zero ... and ...

[Graphics:Images/index_gr_11.gif]
[Graphics:Images/index_gr_12.gif]

... the evident even spacing of the extrema seems confirmed by the next calculation:

[Graphics:Images/index_gr_13.gif]
[Graphics:Images/index_gr_14.gif]
[Graphics:Images/index_gr_15.gif]

Terms 1,000

With one thousand terms, the function looks more like a step function, but the oscillations are still present.

[Graphics:Images/index_gr_16.gif]
[Graphics:Images/index_gr_17.gif]
[Graphics:Images/index_gr_18.gif]
[Graphics:Images/index_gr_19.gif]

Now it seems that there are nine hundred and ninety-nine evenly spaced extrema.

[Graphics:Images/index_gr_20.gif]
[Graphics:Images/index_gr_21.gif]
[Graphics:Images/index_gr_22.gif]

Terms 10,000

As the number of terms is increased without bound, the function converges to the step function.

[Graphics:Images/index_gr_23.gif]
[Graphics:Images/index_gr_24.gif]
[Graphics:Images/index_gr_25.gif]
[Graphics:Images/index_gr_26.gif]
[Graphics:Images/index_gr_27.gif]

Initializations

[Graphics:Images/index_gr_28.gif]

The function with 100 terms is numerically differentiated on the page here.


Converted [in part] by Mathematica      February 18, 2010
Modified: 4 February 2011