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I have a question which I think is very advanced and weird. But I need the answer for some signal analysis purpose.

As we know, the derivative of a sine function, per se, shifts the phase, by Pi/2; i.e.,

f(x) = A sin (w t)

df(x)/dt = A sin (w t + Pi/2) = A cos(w t)

and of course, the integral does the opposite:

Integral(f(x) dx) = A sin(w t - Pi/2) = - A cos(w t).

As we know, there's something called fractional calculus, which involves taking derivates in non-integer orders. So for example the derivative of

g(x) = x

of the order 1/2, would be something like

(d^(1/2) g(x))/(dx^(1/2)) = Gamma(1/2) x^(1/2),

where Gamma(1/2) is the Euler gamma function (I'm not sure whether inside gamma is over 1/2 or 3/2, but whatever).

------------

My question:

So my question is, could you guys help me in finding the phase dependence of sinusoidal functions on non-integer derivatives? so I'm looking for something like:

d^(n) f(x)/dx^n = A sin(w t + k(n)),

where n is a real number, and k(n) is the phase dependence on the derivative order, n, that I'm looking for.

Thank you for any efforts :)

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# Fractional Calculus on Sinusoidal functions

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