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You are not logged in. #1 20130203 20:21:23
Kummer and a tough integral.Hi; This one can resist normal methods quite well. When we say evaluate numerically we typically mean 10  12 digits with a programmable calculator and at least 100 when using a CAS. To start, although any CAS known will gag on this problem they are able to do pieces of so it is to our advantage to split the integral like this. The first integral is not too difficult and any CAS can do it to 100 digits. That is about 105 digits or so. This is the hard part: we notice that asymptotically as x grows large the denominator will act like x^2. The x^2 term will completely drown out the cos(x). That suggests the following chain of moves, Looking at the RHS we see the same idea works again. As x grows large the denominator acts like x^4 completely drowning out the x^2 cos(x) term. So let's add a cos(x) / x^4 to it, we get As x grows larger the denominator acts like x^6 completely drowning out the x^4 cos(x) term. So let's subtract a cos(x)^2 / x^6 to it, we get Two things should be apparent, 1) we can continue the series on the left indefinitely and the RHS is getting smaller and smaller for each term on the left. The RHS will eventually be 0 as we get an infinite series on the LHS. This becomes: What have we accomplished? We have replaced a tough integral with a sum of infinite number of hopefully easier to evaluate numerically integrals. The beauty is that although we have an infinite number of integrals we will only need a finite number of them for the accuracy required. Actually we will only need the first 26 or so terms. Now I am not proposing this method of evaluating that integral because in practice those 26 integrals can also pose problems and further work is needed. But you wanted to see how the acceleration works. 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. #2 20130203 23:03:02
Re: Kummer and a tough integral.How many terms would you need for 1000 digits? The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #3 20130204 00:41:13
Re: Kummer and a tough integral.Hi; 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. #4 20130204 00:46:42
Re: Kummer and a tough integral.Do sum accelerators work on it (like RRA and stuff...) ? The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #5 20130204 00:53:24
Re: Kummer and a tough integral.I do not know the answer to that. I gave up on it when the 26 integrals were more difficult than they should have been. 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. #6 20130204 00:58:35
Re: Kummer and a tough integral.Of course! The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #7 20130204 01:11:26
Re: Kummer and a tough integral.This method was showed to me. now according to him we can interchange the order of the sum and the integral to get: Mathematica can get the sum. Now you just numerically integrate 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. #8 20130204 01:28:40
Re: Kummer and a tough integral.Have you been able to get 1000 digits out of that? The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #9 20130204 01:32:08
Re: Kummer and a tough integral.I have not tried! 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. #10 20130204 01:33:20
Re: Kummer and a tough integral.Why not? The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #11 20130204 01:35:43
Re: Kummer and a tough integral.For several reasons I do not like that solution. 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. #12 20130204 01:40:39
Re: Kummer and a tough integral.What do you not understand? Last edited by anonimnystefy (20130204 01:40:57) The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #13 20130204 01:55:19
Re: Kummer and a tough integral.The first line, beats me. 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. #14 20130204 01:58:41
Re: Kummer and a tough integral.Well, if you do not understand it, there must be a part which you do not understand... The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #15 20130204 02:02:38
Re: Kummer and a tough integral.The trig substitution that produced the series, I do not get. 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. #16 20130204 02:07:30
Re: Kummer and a tough integral.He just splits it up in intervals and uses cos(x+2n*pi)=cos(x). I do not see what the big deal is... The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #17 20130204 02:15:06
Re: Kummer and a tough integral.The cos(x) does not change, the x^2 does. Why? 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. #18 20130204 02:38:04
Re: Kummer and a tough integral.Because cos(x+n*2pi)=cos(x). The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #19 20130204 02:41:49
Re: Kummer and a tough integral.Hi;
I am not getting that. 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. #20 20130204 02:45:01
Re: Kummer and a tough integral.Cosine has a period of 2pi, that's why how ever many times you add 2pi to the argument of cosine you will get the same thing. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #21 20130204 02:47:19
Re: Kummer and a tough integral.I know that one but I still do not undertsand what he did. 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. #22 20130204 02:54:28
Re: Kummer and a tough integral.Hi The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #23 20130204 03:12:24
Re: Kummer and a tough integral.I get a big mess: I left out the intervals of integration because they remain the same. 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. #24 20130204 03:15:48
Re: Kummer and a tough integral.What do you get as the differential of t, i.e. as dt? The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #25 20130204 03:17:24
Re: Kummer and a tough integral.I got dt = 2 π n dx 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. 