Is there some analytical method to calculate the hadamard finite integral, for example , $\int_0^1 |c-x|^{-\alpha} dx$, for $c \in (0,1)$ and alpha not integer, and $1<\alpha<2$. And is there any reference books about Hadamard finite part integrals?
Thanks.
Up to Wiki, we start from the calculation of the integral $$J(\alpha):=\int_0^1 \frac 1 {|c-x|^{\alpha-1}}dx=-{\frac {{c}^{2-\alpha}+ \left( 1-c \right) ^{-\alpha}{c}^{2}-2\, \left( 1-c \right) ^{-\alpha}c+ \left( 1-c \right) ^{-\alpha}}{\alpha -2}}. $$ Then we consider $$\frac{J'(\alpha)} {1-\alpha}={\frac {{c}^{2-\alpha}+ \left( 1-c \right) ^{-\alpha}{c}^{2}-2\, \left( 1-c \right) ^{-\alpha}c+ \left( 1-c \right) ^{-\alpha}}{ ( 1-\alpha) \left( \alpha-2 \right) ^{2}}}-$$ $${\frac {-{c}^{2-\alpha}\ln \left( c \right) - \left( 1-c \right) ^{- \alpha}\ln \left( 1-c \right) {c}^{2}+2\, \left( 1-c \right) ^{- \alpha}\ln \left( 1-c \right) c- \left( 1-c \right) ^{-\alpha}\ln \left( 1-c \right) }{(1-\alpha )\left( \alpha-2 \right) }}$$
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