Series2 performs a series expansion around
0. Series2 is (up to the
Gamma-bug in Mathematica versions smaller than 5.0)
equivalent to Series, except that it applies
Normal on the result and has an option
FinalSubstitutions.
Series2[f, e, n] is equivalent to
Series2[f, {e, 0, n}].
Series2[(x (1 - x))^(\[Delta]/2), \[Delta], 1]\frac{1}{2} \delta \log (1-x)+\frac{1}{2} \delta \log (x)+1
Series2[Gamma[x], x, 1]\frac{1}{2} \zeta (2) x+\frac{1}{x}
Series[Gamma[x], {x, 0, 1}]\frac{1}{x}-\gamma +\frac{1}{12} \left(6 \gamma ^2+\pi ^2\right) x+O\left(x^2\right)
Series2[Gamma[x], x, 2]-\frac{x^2 \zeta (3)}{3}+\frac{1}{2} \zeta (2) x+\frac{1}{x}
Series2[Gamma[x], x, 2, FinalSubstitutions -> {}] // FullSimplify\frac{1}{6} \left(-3 \gamma \left(\zeta (2) x^2+2\right)-2 x^2 \zeta (3)-\gamma ^3 x^2+3 \zeta (2) x+3 \gamma ^2 x+\frac{6}{x}\right)
Series[Gamma[x], {x, 0, If[$VersionNumber < 5, 4, 2]}] // Normal // Expand // FullSimplify\frac{1}{12} \left(-2 \gamma ^3 x^2-\gamma \left(\pi ^2 x^2+12\right)+x \left(\pi ^2-4 x \zeta (3)\right)+6 \gamma ^2 x+\frac{12}{x}\right)
There is a table of expansions of special hypergeometric functions.
Series2[HypergeometricPFQ[{1, OPEm - 1, Epsilon/2 + OPEm}, {OPEm, OPEm + Epsilon}, 1], Epsilon, 1]-\frac{2}{\varepsilon }+\frac{2 m}{\varepsilon }+\frac{1}{2} \varepsilon m \psi ^{(1)}(m)-\frac{\varepsilon \psi ^{(1)}(m)}{2}+1
Series2[HypergeometricPFQ[{1, OPEm, Epsilon/2 + OPEm}, {1 + OPEm, Epsilon + OPEm}, 1], Epsilon, 1]\frac{1}{4} \varepsilon \zeta (2) m+\frac{2 m}{\varepsilon }+\frac{1}{4} \varepsilon m \psi ^{(0)}(m)^2+\frac{3}{4} \varepsilon m \psi ^{(1)}(m)-\frac{1}{2} \varepsilon m S_{11}(m-1)
Hypergeometric2F1[1, Epsilon, 1 + 2 Epsilon, x]
Series2[%, Epsilon, 3]\, _2F_1(1,\varepsilon ;2 \varepsilon +1;x)
-2 \varepsilon ^2 \zeta (2)+2 \varepsilon ^3 \;\text{Li}_3(1-x)+2 \varepsilon ^2 \;\text{Li}_2(1-x)-4 \varepsilon ^3 \;\text{Li}_2(1-x) \log (x)-4 \varepsilon ^3 S_{12}(1-x)-2 \varepsilon ^3 \zeta (2) \log (1-x)+4 \varepsilon ^3 \zeta (2) \log (x)-\frac{1}{6} \varepsilon ^3 \log ^3(1-x)-2 \varepsilon ^3 \log (1-x) \log ^2(x)+\varepsilon ^3 \log ^2(1-x) \log (x)-\frac{1}{2} \varepsilon ^2 \log ^2(1-x)+2 \varepsilon ^2 \log (1-x) \log (x)-\varepsilon \log (1-x)+2 \varepsilon ^3 \zeta (3)+1
There are over 100 more special expansions of {}_2 F_1 tabulated in Series2.m.
The interested user can consult the source code (search for
HYPERLIST).