Testwiki:Reference desk/Archives/Mathematics/2007 October 14

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October 14

Equation from graph data

hello! i have a set of data that i have graphed, what i intend is to hopefully find an equation for it. it is here unfortunately the data is 'loosish' (approximated) and so its not a case of joining the dots and going from there, but the data does have a logarithmic (i think) trend. Can anyone point me in the direction of how to derive an equation from this? sorry if my jargon is a bit lo-brow, its been years since this sorta stuff has occupied any mental real estate of mine. thank you!Boomshanka 02:01, 14 October 2007 (UTC)

The Curve fitting article should get you on your way. Looking at the picture, however, I'm not sure you have enough data points to find a good fit. If you can get more, I'd certainly try that before continuing. risk 02:52, 14 October 2007 (UTC)

Try plotting log x versus log(−y), which might suggest a (linear?) relationship between the two. If that relationship is indeed linear, there is a power law relationship between x and y.  --Lambiam 10:09, 16 October 2007 (UTC)

spot my error

ʃ1/x dx = ln (x) + C

but integrating by parts gives

ʃ1/x dx = x/x - ʃx d/dx(1/x) dx  = x/x - ʃx (-1/x2) dx 

ʃ1/x dx = x/x - ʃx (-1/x2) dx = 1 + ʃx/x2 dx = 1 + ʃ1/x dx 

Where does it go wrong?

I also got ʃ1/(a+x) dx = Sum (n=1 to n=infinity) (x/a+x)ⁿ1/n

this is fine for a=1 with x between 0 and 1 - so no problems calculating logs - but when a=0 a similar problem to above occurs giving ln(x)-ln(1)=Sum (1/n) - Sum (1/n)

It's late and I know there's a mistake here - but if anyone can point out the error (especially in the first example) - I'd appreciate it as I can't see were I've gone wrong. 87.102.19.106 04:13, 14 October 2007 (UTC)

I doesn't go wrong. If I understand correctly you assume that

ʃ1/x dx = 1 + ʃ1/x dx (A)

implies

0 = 1 (B)

and that you thus have a contradiction. This is not the case. In fact, (A) is correct and (B) doesn't follow from (A). The reason for this is that ʃf dx is a notation that doesn't represent a particular primitive of f but rather the family of all the primitives of f, and is therefore not subject to the ordinary laws of arithmetic.

The expression g = ʃf dx is really a shortcut for dg/dx = f. (A) is thus equivalent with 1/x = d1/dx + d(ʃ1/x dx)/dx <=> 1/x = 0 + 1/x <=> 1/x = 1/x <=> 0 = 0. Morana 06:14, 14 October 2007 (UTC)

See Invalid proof#Proof that 0 = 1 --Spoon! 08:45, 14 October 2007 (UTC)

Yes thanks - I realised where I made the error later on but was too tired to come and delete.. also understand any oddities in the second part too.87.102.82.26 17:59, 14 October 2007 (UTC)

Solution of x*cos(x) = c

Is there a closed form solution to x*cos(x) = c ? Or maybe a series solution. How to write all solutions of this equation as a general expression? deeptrivia (talk) 19:01, 14 October 2007 (UTC)

I doubt there is a closed form solution, but their is a series. It uses the Taylor series for cosine. Look up that series, and then multiply it by x, this results in a new series, which is equal to $x\cdot cos(x)$. A math-wiki 00:00, 15 October 2007 (UTC)

I should note that in order to find your solutions, you will to apply the general solution to polynomials, since your polynomial is technically of infinite degree. A math-wiki 00:02, 15 October 2007 (UTC)

I'm not too sure what A math-wiki meant by the general solution to polynomials, however, once you have a series describing c in terms of x, the Lagrange inversion theorem should allow you to find a series of x in terms of c. I am fairly certain that, indeed, no closed form solution in elementary functions exists (though you may be able to do something with the Lambert W function). -- Meni Rosenfeld (talk) 00:28, 15 October 2007 (UTC)

The first few terms, valid for $x\approx 0$, are $x=c+\frac{c^3}{2}+\frac{17c^5}{24}+\frac{961c^7}{720}+\cdots$. For different regions you will have to use different series. -- Meni Rosenfeld (talk) 00:53, 15 October 2007 (UTC)

Question 2: Thanks for your replies. How can I numerically find out the first n solutions? deeptrivia (talk) 02:48, 15 October 2007 (UTC)

That depends... In general I would suggest using Newton's method with different initial points. Choosing the correct points that will guarantee not skipping any root can be tricky, but if you can plot a graph of $x\cos x-c$ it can help. Also, due to the periodicity of the cosine, you can expect roots to be placed in spaces of roughly π - so picking as starting points all $\frac{\pi }{2}k$ (and just $\pi (k+\frac{1}{2})$ for higher k) should allow you to find all roots. -- Meni Rosenfeld (talk) 09:10, 15 October 2007 (UTC)