I wanted to find out what g(x) would be if $g(x)= \frac {d^{sin(x)}}{dx^{sin(x)}} cos(x)$, or using the differintegral operator $g(x)= \Bbb {D}^{sin(x)}cos(x)$. After some research on the internet I learned that $(2πiw)^n \mathcal {F}(f(x))=\mathcal {F}(f^{(n)}(x))$ for any complex number n where w is the transform variable and $\mathcal {F}$ is the fourier transform operator. Thusly, $g(x)=\mathcal{F}^{-1}((2πiw)^{sin(x)} \mathcal {F}(f(x)))$. Expanding out I got the equation $$g(x)=\int_{-∞}^{∞}(2πiw)^{sin(x)}\left(\int_{-∞}^{∞}cos(x)e^{-2iπwx}dx\right)e^{2iπwx}dw$$ This is unfortunately where I got stuck. I am not very familiar with improper partial integrals with complex arguments and I couldn't find anyone in my school's math or physics departments who could help me. I tried using Wolfram Alpha but it either didn't properly interpret what I was putting in or returned exactly what I had put in without any alteration. If someone could tell me how to solve this or inform me of an online calculator that will allow me to solve this that would be great. If there is a better way of solving this please inform me.
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There's variable abuse going on in here. The $x$ in the $n=\sin(x)$ part is not being differentiated, so it's a completely different $x$ than the $x$ in the $\cos(x)$. You should use a different variable, and that should help clarify some parts. – Simply Beautiful Art Feb 11 '19 at 00:04
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Noted and corrected. – user245903 Feb 12 '19 at 19:12