Wikipedia:Reference desk/Archives/Mathematics/2011 November 19

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November 19

Differentiability

Define

$f(x):=\left\{{\begin{array}{ll}\cos \left({\frac {1}{x}}\right)&x\neq 0\\0&x=0\end{array}}\right.$

and

$F(x):=\int _{0}^{x}f(y)\,\operatorname {d} \!y.$

Is F differentiable at 0? My gut says no, but unsure of how to establish it. Sorry for the sloppy TeX. 1.41.7.35 (talk) 13:32, 19 November 2011 (UTC)

(I've fixed the TeX for you — Fly by Night (talk) 19:45, 19 November 2011 (UTC))

F is in fact differentiable, and $F'(0)=0$. Currently I know this only by numerical observation, but to show this formally you'll probably want to use the definition of derivative and the substitution $u=1/x$ to make the integral more manageable. -- Meni Rosenfeld (talk) 16:48, 19 November 2011 (UTC)

I tried integrating using Maple and got

$F(x)=x\cos({\frac {1}{x}})+\int _{0}^{1/x}{\frac {\sin(t)}{t}}dt+c$

The second integral is the Sine integral (see Trigonometric integral) Si(1/x) and

$c=-\int _{0}^{\infty }{\frac {\sin(t)}{t}}dt.$

Since F(0)=0 we have:

$F^{\prime }(0)=\lim _{h\to 0}{\frac {F(h)-F(0)}{h}}$

$F^{\prime }(0)=\lim _{h\to 0}{\frac {F(h)}{h}}$

$F^{\prime }(0)=\lim _{h\to 0}\cos({\frac {1}{h}})-{\frac {1}{h}}\int _{1/h}^{\infty }{\frac {\sin(t)}{t}}dt$

Not sure this helps at all... 77.86.108.27 (talk) 19:06, 19 November 2011 (UTC)

Note that if we break f into "lobes", F is an alternating series. Also, the norm of the sum of an alternating series is at most the norm of the first term. So for ${\frac {2}{\pi (2n+3)}}\leq x<{\frac {2}{\pi (2n+1)}}$, we know $|F(x)|\leq \left|\int _{\frac {2}{\pi (2n+3)}}^{\frac {2}{\pi (2n+1)}}\cos(1/t)\operatorname {d} t\right|<{\frac {2}{\pi (2n+1)}}-{\frac {2}{\pi (2n+3)}}\sim {\frac {1}{\pi n^{2}}}\sim {\frac {\pi x^{2}}{4}}$. Using this, it's easy to see that the difference quotient goes to 0.--121.74.125.249 (talk) 21:27, 19 November 2011 (UTC)

I think I misunderstood the function definition when I commented earlier (I didn't appreciate that f(0) was 'defined' to be zero. Still if you plot

$\int _{0}^{x}\cos(1/t)dt$

it is discontinuous at the origin: approaching from 0+ the function is near the origin, whereas from 0- it's near $-\pi$.

${\text{Let }}G(x)=\int _{0}^{x}\cos(1/t)dt$

$G(x)-G(-x)=\int _{-x}^{x}\cos(1/t)dt$

$G(x)-G(-x)=2x\cos(1/x)+\int _{-1/x}^{1/x}{\frac {\sin(t)}{t}}dt$

So $G(0)-G(-0)=\int _{-\infty }^{\infty }{\frac {\sin(t)}{t}}dt=Pi$ — Preceding unsigned comment added by 77.86.108.27 (talk) 10:15, 20 November 2011 (UTC)

Something is wrong with your plotting. Cosine is even, so F(x) is odd.--121.74.125.249 (talk) 10:34, 20 November 2011 (UTC)

Obviously $\lim _{x\to 0^{-}}F(x)=0$. The mistake (which Mathematica did for me, and probably Maple did for you) is that when you try to express the function with a sine integral, the constant is different for positive or negative x, and the generated expression assumes positive x. Anyway, since the function was defined only for nonnegative x it is understood that we are after the right-derivative, though the derivative is still defined and 0 if you consider both positive and negative x.

Ah okay, so F(x) is (implicitly) defined only for non-negative x, because for negative x we'd want:

$F(x)=\int _{x}^{0}f(y)dy$ 77.86.108.27 (talk) 13:43, 20 November 2011 (UTC)

Actually, I was in error - I meant that f was only defined for $x\geq 0$, but looking again it was actually defined for all x. So F is defined for all x too, because by convention $\int _{a}^{b}f(t)\ dt=-\int _{b}^{a}f(t)\ dt$ for $b<a$.

The point is valid, though - in this light it is still clear that $\lim _{x\to 0}F(x)=0$. -- Meni Rosenfeld (talk) 18:47, 20 November 2011 (UTC)

Also, the value of f at 0 is completely irrelevant to the question. -- Meni Rosenfeld (talk) 11:04, 20 November 2011 (UTC)

Self-similarity

The logarithmic spiral appears often in nature, as the article can attest to. This spiral has the property of self-similarity, which, I'm guessing, plays a role in why it emerges often in nature. Is there a way of "seeing" why nature would tend to be self-similar? 74.15.139.235 (talk) 22:02, 19 November 2011 (UTC)

That is a very good question and this answer is not final. You should look at it the other way round. There is no need for a reason for something to be self-similar, but there must be a reason for it not being self-similar. In that case it must have some characteristic length. (size, diameter, period ...). If the mechanism for creating or developing the item does not define a characteristic length, then the item will be self-similar. Bo Jacoby (talk) 23:33, 19 November 2011 (UTC).

Thank you, your answer has given me a lot to think about! 74.15.139.235 (talk) 00:10, 20 November 2011 (UTC)

To expand on that thought a bit, imagine your circulatory system. It has veins, arteries and capillaries of various sizes. If your DNA contained separate code for the design of each segment, you would need far more DNA. The only way it can work, then, is to have one basic design, and alter it the fit the situation. The same is true of many other biological systems. StuRat (talk) 00:21, 20 November 2011 (UTC)

Woah. 74.15.139.235 (talk) 07:01, 22 November 2011 (UTC)

Benoit Mandelbrot wrote extensively on self-similarity in nature. His book The Fractal Geometry of Nature is very interesting, and accessible to non-mathematicians. Logarithmic spirals are comparatively simple, though: they tend to arise in systems where the rate of growth is proportional to the current size. Looie496 (talk) 17:48, 20 November 2011 (UTC)

Thanks, I'll be sure to take a look at it. 74.15.139.235 (talk) 07:01, 22 November 2011 (UTC)