I found this formula for the case of operators in H.O.Fattorini's book Second Order Linear Differential Equations in Banach Spaces. page 51.$$(-a)^\alpha=\frac{\sin(\alpha \pi)}{\pi} \int_0^\infty \frac{x^{\alpha-1}}{x-a} (-a) \,dx.$$ Can anyone give me an idea from where this formula came in the scalar case.
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There are several proofs in the answers to this question. – Robert Lee Jun 26 '22 at 21:25
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@RobertLee thanks! that is very helpful. – Hidda Walid Jun 26 '22 at 21:40
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@FrankW yes it does, thanks – Hidda Walid Jun 29 '22 at 10:46