On fractional derivatives of trigonometric functions and exponential functions

In some places, fractional derivative is abbreviated as Dα, dαdtα(ordαdxα), where α is positive but not an integer. The "formulas" dαsin⁡tdtα=sin⁡(t+πα2), dαcos⁡tdtα=cos⁡(t+πα2), and dαeλtdtα=λαeλt are sometimes used. However, they generally hold for α∈Z+={0,1,2,…}. In this paper, we give the exact expressions of fractional derivatives for trigonometric functions and exponential functions.