User:Z E U S/continuousintegralsum

$$ \int _{0}^{1}\mathop{\rm {\frac {d^{\alpha}}{d{t}^{\alpha}}}}f \left(\mathop{\rm t}\right)\mathop{\rm }d \alpha =\int _{-\infty }^{\infty }\int _{0}^{\infty }\frac{f \left(\tau \right)\left(\left(u -1\right)\left(t -\mathop{\rm  }\tau \right)^{u -2}-\left(t -\tau \right)^{u -1}\right)}{\Gamma \left(u \right)}d u \mathop{\rm  }d \tau \ $$

$$ \int _{0}^{1}\!{\frac {d^{\alpha}}{d{t}^{\alpha}}}f \left( t \right) {d\alpha}=\int _{-\infty }^{\infty }\!\int _{0}^{\infty }\!{\frac {f \left( \tau \right) \left(  \left( u \left( t-\tau \right) -1 \right) ^{u-2}- \left( t-\tau \right) ^{u-1} \right) }{\Gamma  \left( u \right) }}{du}{d\tau}+f \left( 0 \right)  \left( \int _{0}^{t}\!\int _{0}^{\infty }\!{\frac {{\tau}^{u-1}}{\Gamma  \left( u \right) }}{du}\,{d\tau}-\int _{0}^{\infty }\!{\frac {{t}^{u-1}}{\Gamma  \left( u \right) }}{du} \right) \ $$