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Appendix A Tables

[page 59, §1]

[59.1.1] Let \alpha\in\mathbb{C}, x>a

 
\displaystyle f(x) \displaystyle(\mathrm{I}_{{a+}}^{{\alpha}}f)(x)
 
\displaystyle f(\lambda x) \displaystyle\lambda^{{-\alpha}}(\mathrm{I}_{{\lambda a+}}^{{\alpha}}f)(\lambda x),\lambda>0 (A.1)
(A.2)
\displaystyle(x-a)^{\beta} \displaystyle\frac{\Gamma(\beta+1)}{\Gamma(\alpha+\beta+1)}(x-a)^{{\alpha+\beta}} (A.3)
\displaystyle\mathrm{Re}\,{\beta}>0
(A.4)
\displaystyle\mathrm{e}^{{\lambda x}} \displaystyle\mathrm{e}^{{\lambda a}}(x-a)^{\alpha}\mathrm{E}_{{1,\alpha+1}}(\lambda(x-a)) (A.5)
\displaystyle\lambda\in\mathbb{R}
(A.6)
\displaystyle(x-a)^{{\beta-1}}\;\mathrm{e}^{{\lambda x}} \displaystyle\frac{\Gamma(\beta)\mathrm{e}^{{\lambda a}}}{\Gamma(\alpha+\beta)}(x-a)^{{\alpha+\beta-1}}_{1}F_{1}(\beta;\alpha+\beta;\lambda(x-a)) (A.7)
\displaystyle\mathrm{Re}\,{\beta}>0
(A.8)
\displaystyle(x-a)^{{\beta-1}}\;\log(x-a) \displaystyle\frac{\Gamma(\beta)(x-a)^{{\alpha+\beta-1}}}{\Gamma(\alpha+\beta)}[\psi(\beta)-\psi(\alpha+\beta)+\log(x-a)] (A.9)
(A.10)
\displaystyle(x-a)^{{\beta-1}}\mathrm{E}_{{\gamma,\beta}}((x-a)^{\gamma}) \displaystyle(x-a)^{{\alpha+\beta-1}}\mathrm{E}_{{\gamma,\alpha+\beta}}((x-a)^{\gamma}) (A.11)
\displaystyle\mathrm{Re}\,{\beta}>0,\mathrm{Re}\,{\gamma}>0
 
Table A.1: Some fractional integrals