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You are not logged in. #26 20121031 01:27:11
Re: Fourier Series for e^xYes, I understand. I thought both diagrams on the right were representing the same thing, which confused me. #27 20121031 01:34:16
Re: Fourier Series for e^xThe circles are my drawings on top of the lines. We do not need any rigor here just an intuitive feel that as the lines get closer and closer the point of intersection becomes fuzzier. For instance if all three drawings were on graph paper it would be easy to read off the point of intersection for the first one with better accuracy than the third one. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #28 20121031 01:44:18
Re: Fourier Series for e^xI see. So the circle gets bigger as the lines get closer. #29 20121031 01:53:50
Re: Fourier Series for e^xYes, it represents a certain amount of error in the calculation. Like your eyes, algorithms that try to find the points of intersection of the third example will experience instability and have more round off error and consequently give poorer results. So lines that cross at nearly right angles are more accurate in a sense than lines that cross at low angles. That is the point. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #30 20121031 01:57:27
Re: Fourier Series for e^xAh, I see. #31 20121031 02:09:06
Re: Fourier Series for e^xWhen 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. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #32 20121031 02:22:08
Re: Fourier Series for e^xI 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? #33 20121031 02:39:27
Re: Fourier Series for e^xHi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #34 20121031 02:41:39
Re: Fourier Series for e^xI see them. So the Fourier series have an orthogonal basis? And I am guessing Taylor series do not? #35 20121031 02:51:00
Re: Fourier Series for e^xTaylor series are not orthogonal but they are osculating so they have some benefits. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #36 20121031 02:55:00
Re: Fourier Series for e^xOh, 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? #37 20121031 02:57:36
Re: Fourier Series for e^xThat I do not know for sure. The Fourier fit is also least squares or minimax, I am not sure. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #38 20121031 03:07:02
Re: Fourier Series for e^xWhat do you mean by least squares? I have heard the term thrown around for regression lines in statistics. #39 20121031 03:10:08
Re: Fourier Series for e^xLeast squares minimize the square of the error between the fit equation and the data. First you start with an overdetermined system. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #40 20121031 03:15:23
Re: Fourier Series for e^xOh okay... but, how can we get this from the Fourier series? #41 20121031 03:19:08
Re: Fourier Series for e^xIt happens automatically. It is inherent in the fit. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #42 20121031 03:21:15
Re: Fourier Series for e^xOkay... #43 20121031 03:27:07
Re: Fourier Series for e^xMost 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! In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #44 20121031 03:36:46
Re: Fourier Series for e^xI 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. #45 20121031 03:43:54
Re: Fourier Series for e^xThat is always a danger and some of that will definitely happen. Everyone has slowed down with mental calculation. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. 