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.

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:

Schröder's equation is also interestingly related

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...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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