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Table 1 Some useful identities of the local fractional calculus are give below

From: Local fractional homotopy analysis method for solving non-differentiable problems on Cantor sets

  Basic identities
1 \(\cos _{\alpha }(t^{\alpha })=\sum_{n=0}^{+\infty }(-1)^{n}\frac{t^{(2n+1)\alpha }}{\varGamma (1+(2n+1)\alpha)}\), 0<α ≤ 1
2 \(\sin _{\alpha }(t^{\alpha })=\sum_{n=0}^{+\infty }(-1)^{n}\frac{t^{2n\alpha }}{\varGamma (1+(2n+1)\alpha)}\), 0<α ≤ 1
3 \(E_{\alpha }(t^{\alpha })=\sum_{n=0}^{+\infty }\frac{t^{n\alpha }}{\varGamma (1+n\alpha)}\), 0<α ≤ 1
4 \(\frac{d^{\alpha }}{dt^{\alpha }}\frac{t^{n\alpha }}{\varGamma (1+n\alpha)}=\frac{t^{(n-1)\alpha }}{\varGamma (1+(n-1)\alpha)}\)
5 \({}_{0}I^{(\alpha)}_{t}\frac{t^{n\alpha }}{\varGamma (1+n\alpha)}=\frac{t^{(n+1)\alpha }}{\varGamma (1+(n+1)\alpha)}\)
6 \(\frac{d^{\alpha }}{dt^{\alpha }}\cos _{\alpha }(t^{\alpha })=-\sin _{\alpha }(t^{\alpha })\)
7 \(\frac{d^{\alpha }}{dt^{\alpha }}\sin _{\alpha }(t^{\alpha })=\cos _{\alpha }(t^{\alpha })\)
8 \(\frac{d^{\alpha }}{dt^{\alpha }}E_{\alpha }(t^{\alpha })=E_{\alpha }(t^{\alpha })\)