Fractional powers of the derivative operator are still linear operators: the usual derivative sends e^kx to ke^kx; the fractional (d/dx)^a sends e^kx to (k^a)e^kx. Then you can build other functions out of Fourier modes if k is complex.

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In fact it's not that simple because k^a is ambiguous when k is complex and a is fractional, this leads to multiple definitions. See: sciencedirect.com/science/arti

Schröder's equation is also interestingly related en.wikipedia.org/wiki/Schr%C3%

...and you can even have one on vector fields. This whole thread was caused by me finding √−∇² in a formula

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