Wikipedia:Reference desk/Archives/Mathematics/2011 February 22

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February 22

integrating

integrating

${\frac {d\ln y_{1}}{dt}}=a_{1}-b_{1}y_{2}$

between 0 and T.

Can I write

$\int _{0}^{T}d\ln y_{1}=\int _{0}^{T}a_{1}-b_{1}y_{2}dt$

to get

$\ln {\frac {y_{1}(T)}{y_{1}(0)}}=a_{1}T-b_{1}\int _{0}^{T}dy_{2}dt$

The middle line has no "dt" on the left hand side so I'm not sure if this is right. —Preceding unsigned comment added by 130.102.158.15 (talk) 04:06, 22 February 2011 (UTC)

It's correct, though depending on how you define things it can be considered an abuse of notation. There's a lot of mileage you can get from abusing the differential notation. -- Meni Rosenfeld (talk) 09:35, 22 February 2011 (UTC)

Is there a way to do it without abusing notation? —Preceding unsigned comment added by 130.102.158.15 (talk) 09:42, 22 February 2011 (UTC)

Sure. Just substitute $u=\ln y_{1}$, and you get $u'(t)=a_{1}-b_{1}y_{2}(t)\;\!$ and so $u(T)=u(0)+\int _{0}^{T}(a_{1}-b_{1}y_{2}(t))dt$, hence $\ln {\frac {y_{1}(T)}{y_{1}(0)}}=u(T)-u(0)=a_{1}T-b_{1}\int _{0}^{T}y_{2}(t)dt$.

By the way, I've just noticed there's a stray d on your RHS. -- Meni Rosenfeld (talk) 11:05, 22 February 2011 (UTC)

Another way to look at it without invoking any substitutions is that you could (should?) go from the first equation to the second one via this intermediate step:

$\int _{0}^{T}{\frac {d\ln y_{1}}{dt}}dt=\int _{0}^{T}a_{1}-b_{1}y_{2}dt$

In this sense you "integrate both sides w.r.t. t from 0 to T", which is the proper way to do it. Then apply the fundamental theorem of calculus to end up with $\ln {\frac {y_{1}(T)}{y_{1}(0)}}$ on the LHS. Zunaid 11:32, 22 February 2011 (UTC)

Remember the parentheses, and specify which variable the limits refer to:

$\int _{t=0}^{T}d\ln y_{1}=\int _{0}^{T}(a_{1}-b_{1}y_{2})dt$

to get

$\ln {\frac {y_{1}(T)}{y_{1}(0)}}=a_{1}T-b_{1}\int _{0}^{T}y_{2}dt$

Bo Jacoby (talk) 12:46, 22 February 2011 (UTC).

The OP has

$\int _{0}^{T}dy_{2}\,dt$

where

$\int _{0}^{T}y_{2}\,dt$

should be. Michael Hardy (talk) 22:59, 23 February 2011 (UTC)

Functional square roots

I recently found a Wikipedia article about functional square roots, i.e. functions "halfway between" a function. To be more precise, the functional square root of a function f is a function g so that g(g(x))=f(x). Now I have not studied this very much, but my intuition says that not nearly all possible functions have functional square roots at all. Is this correct? What sort of functions have functional square roots? JIP | Talk 19:37, 22 February 2011 (UTC)

Having read this article a couple of years ago myself I was also interested in questions such as these. I think the length of the article kind of gives it away: this doesn't seem to be a fruitful area of mathematical research and thus results are sparse and obscure. Throwing the necessary terms at Google gives the following interesting links to more discussion on the topic: solving f(f(x))=g(x) at MathOverflow and this thread at Google Answers. I'd be keen to here opinions from the real mathematicians on this board though. Perhaps User:Dmcq will pop around, he was active on the talk page of that article in 2010. Zunaid 21:22, 22 February 2011 (UTC)

Generators of functions

Related to the above question on functional square roots, one can also consider the functional equation:

g(g(g(....g(x)....))) = f(x)

where f(x) is given and we iterate some function g(x), n times. In the large n limit, one would expect to get:

g(x) = 1 + 1/n h(x) + O(1/n^2)

and h(x) can be called a generator of f(x). I know that in conformal field theory one studies algebraic relations involving generators, but I have never seen an explicit expressions for a generator corresponding to some given analytic function. Count Iblis (talk) 23:57, 22 February 2011 (UTC)

Well I amused myself with this idea some time ago and I thought the most interesting ones had g(0)=0. So I considered things like for instance e^x-1 rather than just e^x. You can get a nice equation for 'generators' for things like this with x+x²h(x)/n and I was wondering what happened when one combined two generators together for instance. Nothing all that interesting that I could see unfortunately. In general the 'generator' won't have a nice convergent Taylor's series even if the original function does - one can calculate what the terms would be if it was okay but usually they diverge. I didn't read up on it at the time and only looked up enough later to add a couple of citations to the article which was citationless. Dmcq (talk) 00:48, 23 February 2011 (UTC)