∫ 0 1 d α d t α f ( t ) d α = ∫ − ∞ ∞ ∫ 0 ∞ f ( τ ) ( ( u − 1 ) ( t − τ ) u − 2 − ( t − τ ) u − 1 ) Γ ( u ) d u d τ {\displaystyle \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)}}du\mathop {\rm {}} d\tau \ }
∫ 0 1 d α d t α f ( t ) d α = ∫ − ∞ ∞ ∫ 0 ∞ f ( τ ) ( ( u ( t − τ ) − 1 ) u − 2 − ( t − τ ) u − 1 ) Γ ( u ) d u d τ + f ( 0 ) ( ∫ 0 t ∫ 0 ∞ τ u − 1 Γ ( u ) d u d τ − ∫ 0 ∞ t u − 1 Γ ( u ) d u ) {\displaystyle \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)\ }
