Wikipedia:Reference desk/Archives/Mathematics/2011 October 27

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October 27

Is this an abuse of notation?

$\int _{0}^{x}f(x)dx$ using x as the dummy variable and the limit.Widener (talk) 02:00, 27 October 2011 (UTC)

Yes, it is; but anyone who understands that the integration variable is a dummy variable can figure out what it means-- though they'll probably want you to have a good reason for not using "t" or something else as the dummy variable. Those who don't understand dummy variables will probably interpret it as you intend, but you're not doing them any favors by using confusing notation.

I just recently saw this abuse of notation in a seminar talk presenting new research, where the presenter wanted to avoid using too many variables. There was an expression like $f(x_{1},x_{2},\ldots ,x_{k})=\left(\int _{0}^{x_{1}}{\frac {dx_{1}}{x_{1}}}\right)^{n_{1}-1}\int _{0}^{x_{1}}{\frac {dx_{1}}{1-x_{1}}}\left(\int _{0}^{x_{2}}{\frac {dx_{2}}{x_{2}}}\right)^{n_{2}-1}\int _{0}^{x_{2}}{\frac {dx_{2}}{1-x_{2}}}\cdots \left(\int _{0}^{x_{k}}{\frac {dx_{k}}{x_{k}}}\right)^{n_{k}-1}\int _{0}^{x_{k}}{\frac {dx_{k}}{1-x_{k}}}$. He had to stop and explain that these are actually iterated integrals, evaluated from the right to the left. Once he explained what it meant, everyone accepted and understood it. I wondered briefly if there is a better notation; but he was viewing each integral as an operator, and used his notation to suggest this perspective. You can always abuse notation if you have a good reason; there's a long tradition of it in mathematics. But be prepared to defend your choice, and be sure it's clear exactly what you mean. — Preceding unsigned comment added by 173.20.199.11 (talk) 03:12, 27 October 2011 (UTC)

Without explanation, though, it's hard to be sure whether (say) the x₁ in the denominator of the second integral is the variable of integration, or the x₁ on the left-hand side. The usual convention in programming languages is that a variable in a smaller scope "hides" a variable of the same name in a larger scope, so that would argue for the first option.

That's fine for compilers, not so good for humans, and in fact a lot of compilers will throw a warning when this happens, so you're not caught by surprise. --Trovatore (talk) 06:32, 27 October 2011 (UTC)

Isn't

$\left(\int _{0}^{x_{1}}{\frac {dx_{1}}{x_{1}}}\right)^{n_{1}-1}\int _{0}^{x_{1}}{\frac {dx_{1}}{1-x_{1}}}\left(\int _{0}^{x_{2}}{\frac {dx_{2}}{x_{2}}}\right)^{n_{2}-1}\int _{0}^{x_{2}}{\frac {dx_{2}}{1-x_{2}}}\cdots \left(\int _{0}^{x_{k}}{\frac {dx_{k}}{x_{k}}}\right)^{n_{k}-1}\int _{0}^{x_{k}}{\frac {dx_{k}}{1-x_{k}}}$

equal to

$\left(\int _{0}^{x_{1}}{\frac {dt}{t}}\right)^{n_{1}-1}\int _{0}^{x_{1}}{\frac {dt}{1-t}}\left(\int _{0}^{x_{2}}{\frac {dt}{t}}\right)^{n_{2}-1}\int _{0}^{x_{2}}{\frac {dt}{1-t}}\cdots \left(\int _{0}^{x_{k}}{\frac {dt}{t}}\right)^{n_{k}-1}\int _{0}^{x_{k}}{\frac {dt}{1-t}}$

?

If not I don't understand what it means. Bo Jacoby (talk) 11:00, 27 October 2011 (UTC).

The expression $\left(\int _{0}^{x}{\frac {dx}{x}}\right)^{2}\int _{0}^{x}{\frac {dx}{1-x}}$ should be interpreted as $\int _{0}^{x}\int _{0}^{t_{1}}\int _{0}^{t_{2}}{\frac {dt_{3}}{1-t_{3}}}{\frac {dt_{2}}{t_{2}}}{\frac {dt_{1}}{t_{1}}}$, where the integrals are evaluated from the inside out. In this way, you get an iterated integral expression for the polylogarithm. This was the explained in the seminar, and I should have explained it all above. I think I must have some details wrong above, because the original multiple variable expression above is supposed to give a multiple polylogarithm, but it isn't quite right.129.186.252.35 (talk) 15:16, 27 October 2011 (UTC)

compute incomplete beta function

how do you compute the incomplete beta function for x values in the closed range [0, 1] with a=0 and b=0 or either of them being 1? thanks in advance!218.103.211.241 (talk) 13:35, 27 October 2011 (UTC)

Call the function from GSL to compute incomplete Beta, call the function from Mathematica , call the function in your handheld calculator, do some numerical approximation (quadrature) for the integration in the defintion, follow the references for tables in DLMF. – b_jonas 11:52, 28 October 2011 (UTC)

Wait, I forgot one: call the function in matlab or octave. – b_jonas 11:54, 28 October 2011 (UTC)

Hm wait, you ask specifically for a=b=0 and simple values like that? That could be easier. Try to write the definition and compute the indefinite integral explicitly. – b_jonas 11:57, 28 October 2011 (UTC)

Thanks, ill be trying that out.218.103.211.241 (talk) 13:07, 28 October 2011 (UTC)

Generate values for a "logarithmic sweep"

Hi all,

Background context: I am using a circuit simulator application that includes a frequency generator. When setting up a simulation, I can define a sweep of frequencies within a range, and, in particular, I can tell it to give me a logarithmic sweep with n steps. What I am trying to figure out is how these values are chosen.

If I ask my application for a logarithmic sweep between 1000 and 2000 in 10 steps, the application gives me back

1000.00, 1080.06, 1166.53, 1259.92, 1360.79, 1469.73, 1587.40, 1714.49, 1851.75, 2000.00.

I want to figure out the formula that will generate these values. Doing some research, I found the formula f(x) = fStart + 10^(log(range)/steps * x). This gave me what seemed to me naively to be a logarithmic sweep. Each step is twice that of the step before (so the last three values in the series from 1000 to 2000 were 1250, 1500 and 2000). However, this is clearly different than the numbers above. Further, if I plot the two series on a log scale, the series above gives me a straight line, while my formula does not. Clearly the application's sweep is correct, and my formula is something else.

So, any ideas on how to produce a "log sweep" that will give me the series above? Thanks! — Sam 66.31.201.89 (talk) 21:44, 27 October 2011 (UTC)

The above values have the property the ratio between any consecutice pair is the 9th root of 2, i.e. 1.08005974..., so the ith one will be the start value multiplied by the ith power of this.←86.155.185.195 (talk) 22:23, 27 October 2011 (UTC)

(ec)

$1000\cdot 2^{\frac {n}{9}}$ for n=0,1,2,3,4,5,6,7,8,9. Bo Jacoby (talk) 22:26, 27 October 2011 (UTC).

66.31, note two things when comparing your frequency generator's numbers and your formula. First, when you mentioned "ten steps", you need to be careful if you are counting the fenceposts or fencerails. Your frequency generator's ten values grow in nine steps, thus the 9th root mentioned by 86.155 and Jacoby. Second, your formula, f(x) = fStart + 10^(log(range)/steps * x), simplifies to f(x) = fStart + range^(x/steps). It does represent exponential growth, but from an initial offset. -- 110.49.233.75 (talk) 05:10, 28 October 2011 (UTC)

Ok, I see that your formula works, but I'm trying to generalize it to a formula for any starting and ending values, and step size and not succeeding. Your formula above looked at first glance to be simply $start\cdot 2^{\frac {n}{steps-1}}$, but obviously this isn't it, because the end isn't used anywhere. How would I calculate the steps for, say, a log sweep between 150 and 200 in 5 steps (the answer of which, by the way, is 150, 161 173 186, 200). Thanks, — Sam 66.31.201.89 (talk) 13:00, 28 October 2011 (UTC)

The 2 comes from fEnd / fStart = 2000 / 1000. So, you could write f(x) = fStart * (fEnd / fStart) ^ (x / steps) for x = 0, 1, ... steps-1, steps. Note that the log and exp functions can be used, as in your first equation, in place of the general y^x function, so you could write f(x) = fStart * 10^(log(fEnd / fStart) * x / steps), where log is base 10 logarithm or the 10^ is replaced with exponentiation of the same base as the log. In words, you want to multiply each term of your sequence with a constant multiplier yielding a sequence of the desired length (steps), starting with fStart and finally resulting in fEnd. If you only wanted two terms (one step) you could multiply fStart by (fEnd/fStart) to get (fStart, fEnd). If you wanted three terms (two steps) you could multiply fStart by sqrt(fEnd/fStart) and then multiply that result again by the same value to get (fStart, fStart*sqrt(fEnd/fStart), fEnd) (note that the middle term equals sqrt(fStart * fEnd), or the geometric mean of the endpoints. If you wanted n+1 terms (n steps) you would use the multiplier (fEnd/fStart)^(1/n) or the nth root of the ratio of the endpoints. Does that make sense? (I still think of your 10 values as involving 9 steps. If you wish to call it 10 steps then simply adjust things accordingly.)-- 110.49.233.75 (talk) 14:45, 28 October 2011 (UTC)

Another way to derive it is to say that you want a geometric sequence (a₀, a₁, ... a_n) where a₀ = fStart and a_n = fEnd. Being a geometric sequence, a_k = r * a_k-1 for some common ratio r and thus a_k = r^k * a₀. (Note that I am indexing the sequence from 0, where part of the geometric sequence article indexes from 1.) Thus a_n = rⁿ * a₀, but a₀ = fStart and a_n = fEnd, so fEnd = rⁿ * fStart and rⁿ = fEnd / fStart and r = (fEnd / fStart)^1/n. -- 110.49.233.75 (talk) 15:00, 28 October 2011 (UTC)

I got it! Thank you very much for your answers and explanations. — Sam 76.118.180.196 (talk) 17:37, 28 October 2011 (UTC)