Theory and Modern Applications
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
| Â | 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 | \(\frac{d^{\alpha }}{dt^{\alpha }}E_{\alpha }(t^{\alpha })=E_{\alpha }(t^{\alpha })\) |