A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions

Table 1

Fractional order derivatives for some functions.

Functions	L-fractional derivatives
𝑥^𝛽, 𝛽>−1	𝑥^−𝛼+𝛽Γ1+𝛽Γ1−𝛼+𝛽
Cos(𝑎𝑥), 𝑎∈ℝ	𝑥^−𝛼HypergeometricPFQ{1},{1−𝛼/2,3/2−𝛼/2}−(1/4)𝑎^2𝑥^2−𝑎Γ(1−𝛼)^2𝑥^2−𝛼HypergeometricPFQ{2},{2−𝛼/2,5/2−𝛼/2}−(1/4)𝑎^2𝑥^2Γ(4−𝛼)
Sin(𝑎𝑥), 𝑎∈ℝ	𝑎𝑥^1−𝛼(2−𝛼)HypergeometricPFQ{1},{2−𝛼/2,3/2−𝛼/2}−(1/4)𝑎^2𝑥^2−Γ(2−𝛼)2𝑎^3𝑥^3−𝛼HypergeometricPFQ{2},{5/2−𝛼/2,3−𝛼/2}−(1/4)𝑎^2𝑥^2Γ(5−𝛼)
ln(𝑥)	−𝑥^−𝛼EulerGamma+𝜋Cot[𝜋𝛼]−ln(𝑥)+PolyGamma[0,𝛼]Γ(1−𝛼)
𝑒^𝑎𝑥, 𝑎∈ℝ	𝑎^𝛼(𝑎𝑥)^−𝛼+𝑒^𝑎𝑥(Γ(1−𝛼)−Γ(1−𝛼,𝑎𝑥))Γ(1−𝛼)
Sinh(𝑎𝑥)	𝑒^−𝑎𝑥−𝑎^2𝑥^2−𝛼2Γ(1−𝛼)−𝑎^2𝑥^2𝛼(−𝑎)^𝛼(𝛼Γ(−𝛼)+Γ(1−𝛼,−𝑎𝑥))+𝑎^𝛼𝑒^2𝑎𝑥(Γ(1−𝛼)−Γ(1−𝛼,𝑎𝑥))
Cosh(𝑎𝑥)	1()𝑒2Γ1−𝛼^−𝑎𝑥−𝑎^2𝑥^2−𝛼(−𝑎)^𝛼(𝑎𝑥)^𝛼𝑒^𝑎𝑥+𝑎𝑥ExpIntegral𝐸[𝛼,−𝑎𝑥]+𝑎^𝛼𝑒^𝑎𝑥(−𝑎𝑥)^𝛼1−𝑎𝑒^𝑎𝑥𝑥𝑎𝑥ExpIntegral𝐸[𝛼,𝑎𝑥]+(−𝑎)^𝛼+𝑎^𝛼𝑒^2𝑎𝑥−𝑎^2𝑥^2𝛼Γ(1−𝛼)
Arcsin(𝑥), 0<𝑥<1	𝑥^1−𝛼HypergeometricPFQ{1/2,1/2,1},{3/2−𝛼/2,2−𝛼/2},𝑥^2Γ()+2−𝛼2𝑥^3−𝛼HypergeometricPFQ{3/2,3/2,2},{5/2−𝛼/2,3−𝛼/2},𝑥^2Γ(5−𝛼)
Arccos(𝑥), 0<𝑥<1	−𝑥^−𝛼𝜋(−2+𝛼)+2𝑥HypergeometricPFQ{1/2,1/2,1},{3/2−𝛼/2,2−𝛼/2},𝑥^22()Γ()−𝑥2−𝛼1−𝛼^1−𝛼HypergeometricPFQ{1/2,1/2,1},{3/2−𝛼/2,2−𝛼/2},𝑥^222−3𝛼+𝛼^2Γ()−𝑥1−𝛼3−𝛼HypergeometricPFQ{3/2,3/2,2},{5/2−𝛼/2,3−𝛼/2},𝑥^222−3𝛼+𝛼^2Γ(1−𝛼)((3/2−𝛼/2)(2−𝛼/2))
Arctan(𝑥)	−𝑥^1−𝛼Γ()()5−𝛼−4+𝛼)(−3+𝛼)(−2+𝛼HypergeometricPFQ12,3,1,12−𝛼2𝛼,2−2−𝑥^2+4𝑥^2HypergeometricPFQ32,5,2,22−𝛼2𝛼,3−2,−𝑥^2
∫^∞𝑥𝑒^−𝑦𝑦𝑑𝑦	1Γ()𝑒3−𝛼^−𝑥𝑥^−1−𝛼(−𝑥)^𝛼(−2+𝛼)(Γ(2−𝛼)−Γ(2−𝛼,−𝑥))+𝑒^𝑥𝑥(−1+𝛼)𝑥HypergeometricPFQ[{1,1},{2,3−𝛼},−𝑥](−2+𝛼)𝜋Cot(𝜋𝛼)−ln(𝑥)+PolyGamma(0,𝛼)
𝐸𝛼−𝑡^𝛼	𝑥^−𝛼Γ(1−𝛼)−𝐸𝛼−𝑡^𝛼
𝐽𝑛(𝑥),; Re[𝑛]>−1	2^−𝑛𝑥^𝑛−𝛼(1+𝑛−𝛼)Γ(2+𝑛−𝛼)HypergeometricPFQ12+𝑛2𝑛,1+2,𝑛1+𝑛,1+2−𝛼2,32+𝑛2−𝛼2,−𝑥^24+2^−1−𝑛(1/2+𝑛/2)(1+𝑛/2)𝑥^2+𝑛−𝛼(1+𝑛)(1+𝑛/2−𝛼/2)(3/2+𝑛/2−𝛼/2)Γ(2+𝑛−𝛼)HypergeometricPFQ32+𝑛2𝑛,2+2,𝑛2+𝑛,2+2−𝛼2,52+𝑛2−𝛼2,−𝑥^24
𝐾𝑛(𝑥),; 1> Re[𝑛] >−1	2^−1−𝑛𝜋𝑥^1−𝑛−𝛼Csc()4𝑛𝜋^𝑛Γ(2+𝑛−𝛼)HypergeometricPFQ12−𝑛2𝑛,1−2,𝑛1−𝑛,1−2−𝛼2,32−𝑛2−𝛼2,𝑥^24+𝑥^2𝑛Γ(2+𝑛−𝛼)HypergeometricPFQ12+𝑛2𝑛,1+2,𝑛1+𝑛,1+2−𝛼2,32+𝑛2−𝛼2,−𝑥^24
𝑌𝑛(𝑥),; −1< Re[𝑛] <1	2^−1−𝑛𝑥^1−𝑛−𝛼−Csc(𝑛𝜋)4^𝑛Γ(2+𝑛−𝛼)HypergeometricPFQ12−𝑛2𝑛,1−2,𝑛1−𝑛,1−2−𝛼2,32−𝑛2−𝛼2,−𝑥^24+𝑥^2𝑛Γ(2+𝑛−𝛼)HypergeometricPFQ12+𝑛2𝑛,1+2,𝑛1+𝑛,1+2−𝛼2,32+𝑛2−𝛼2,−𝑥^24
Zeta[𝑥]	^∞∑𝑛=1(−1)^𝛼(−𝑥)^−𝛼Γ()−𝑛1−𝛼^−𝑥𝛼Γ(−𝛼)(−ln(𝑛))^𝛼Γ()−𝑛1−𝛼^−𝑥𝛼Γ(1−𝛼,−𝑥ln(𝑛))(−ln(𝑛))^𝛼Γ(1−𝛼)
Erf(𝑡)	2^−1+𝑥𝑥^1−𝛼(2−𝛼)HypergeometricPFQRegularized12,1,32−𝛼2𝛼,2−2−𝑥^2+2^−1+𝑥𝑥^3−𝛼HypergeometricPFQRegularized32,2,52−𝛼2𝛼,3−2−𝑥^2