Wikipedia:Reference desk/Archives/Mathematics/2014 May 22

= May 22 =

All Solutions for F'(x) = F(x+a).
We know, for instance, that $$\exp'(x)=\exp(x+0),$$ $$\sin'x=\sin\bigg(x+\frac\pi2\bigg),$$ and $$\cos'x=\cos\bigg(x+\frac\pi2\bigg).$$ But since all these are connected through Euler's formula, I was wondering whether there might not be other (non-exponential) functions with the same general property. — 79.118.167.195 (talk) 08:46, 22 May 2014 (UTC)
 * If a is fixed and not zero, then you can recover a solution by imposing basically aribitrary data in the form of a smooth function in [0,a] (with appropriate compatibility conditions at the end points) just by integration, I believe. Sławomir Biały  (talk) 11:35, 22 May 2014 (UTC)
 * Note 'smooth' here has a precise definition, see smooth function. Dmcq (talk) 12:17, 22 May 2014 (UTC)
 * F(x) = 0 is a trivial solution for any a. Gandalf61 (talk) 12:26, 22 May 2014 (UTC)


 * Note that it suffices to consider a = 0 and a = 1. Count Iblis (talk) 14:45, 22 May 2014 (UTC)


 * You also have $$\exp'(x)=\exp(x+2in\pi),$$ for all integer n (including zero). The same function but more values for a. You can obviously add 2n$\pi$ to π/2 in the trigonometric relations too. This suggests there are no more functions as the trigonometric and exponential functions satisfy F'(x) = F(x+a) for so many values.-- JohnBlackburne wordsdeeds 15:04, 22 May 2014 (UTC)
 * I don't really know anything about this equation, but it is a delay differential equation (linear with discrete delay), and that article contains a little bit of general information. —Kusma (t·c) 20:57, 22 May 2014 (UTC)