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Topic review (newest first)
- bobbym
- 2012-10-31 03:43:54
That is always a danger and some of that will definitely happen. Everyone has slowed down with mental calculation.
What is the upside?
1) You are freed up from the drudgery that takes much time.
2) You can think about the problem without getting sidetracked with the details of solving it.
3) By getting passed things you could not get passed before you now can delve into deeper mathematics because you can use your package to help read journal articles. Incidentally, they do not admit it in the article but most journal articles are now written with the help of a CAS.
4) And this is a big one you never make a careless mistake that blows a solution away. For instance you might have wondered about that faulty solution for many months.
- zetafunc.
- 2012-10-31 03:36:46
I am still learning though. For instance, I did not want to just type in "FourierSeries[e^x]" because that won't show me how they dealt with the problem I had in post #1, which was having an undefined term (division by 0) at n = 1. The worry I have, for example, is having my knowledge of the concept deteriorate. Now that I use a calculator quite often I do not have to do something like 357*762 in my head, so over time, I have got slower at doing mental calculations. I am worried that the same thing would happen with this, for example. I may forget how to find Fourier series because I am used to getting something to do it for me.
- bobbym
- 2012-10-31 03:27:07
Most of the time I use Mathematica. Today algebra is sort of like square roots of numbers. If you had to evaluate √ (234.176253) you would turn to your calculator. If you needed to multiply 102536 * 776241 you would turn to your calculator. At one time people did them both with pencil and paper. Same thing now with mechanical symbolic math. When you are learning do it by hand, when you know it use a CAS!
- zetafunc.
- 2012-10-31 03:21:15
Okay...
By the way, when you derived that Fourier series for e^x, did you use Mathematica or did you derive it yourself? When I did it myself, I did not appear to get the right answer. Should I post my working?
- bobbym
- 2012-10-31 03:19:08
It happens automatically. It is inherent in the fit.
- zetafunc.
- 2012-10-31 03:15:23
Oh okay... but, how can we get this from the Fourier series?
- bobbym
- 2012-10-31 03:10:08
Least squares minimize the square of the error between the fit equation and the data. First you start with an overdetermined system.
- zetafunc.
- 2012-10-31 03:07:02
What do you mean by least squares? I have heard the term thrown around for regression lines in statistics.
- bobbym
- 2012-10-31 02:57:36
That I do not know for sure. The Fourier fit is also least squares or minimax, I am not sure.
- zetafunc.
- 2012-10-31 02:55:00
Oh, I see. Are they orthogonal because sine is 90° out of phase with cosine, and the Fourier series is a sum of sines and cosines?
- bobbym
- 2012-10-31 02:51:00
Taylor series are not orthogonal but they are osculating so they have some benefits.
Now in the plot of e^x and the Fourier series you will not see the orthogonal basis in the plot itself. They are present in the formation of the series.
- zetafunc.
- 2012-10-31 02:41:39
I see them. So the Fourier series have an orthogonal basis? And I am guessing Taylor series do not?
- bobbym
- 2012-10-31 02:39:27
Hi;
When we use an orthogonal basis we get this plot. Notice the high angles of intersection.
- zetafunc.
- 2012-10-31 02:22:08
I understand that orthogonality is preferred since it gives you the least possible error. But I can't see how this relates to our Fourier series for e^x. Where are the orthogonal lines?
- bobbym
- 2012-10-31 02:09:06
When we curve fit using ordinary polynomials x, x^2, x^3, x^4, x^5,...as the basis we can see by graphing how much they are like example 3. Look at that mess around the origin. All of them on top of each other. That is why it is not recommended to curve fit a function using powers higher than say 10. The accumulated error makes them very difficult to get accurate results.