FeynCalc manual (development version)

Series2

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}].

See also

Overview, Series3.

Examples

Series2[(x (1 - x))^(\[Delta]/2), \[Delta], 1]

12δlog(1x)+12δlog(x)+1\frac{1}{2} \delta \log (1-x)+\frac{1}{2} \delta \log (x)+1

Series2[Gamma[x], x, 1]

12ζ(2)x+1x\frac{1}{2} \zeta (2) x+\frac{1}{x}

Series[Gamma[x], {x, 0, 1}]

1xγ+112(6γ2+π2)x+O(x2)\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]

x2ζ(3)3+12ζ(2)x+1x-\frac{x^2 \zeta (3)}{3}+\frac{1}{2} \zeta (2) x+\frac{1}{x}

Series2[Gamma[x], x, 2, FinalSubstitutions -> {}] // FullSimplify

16(3γ(ζ(2)x2+2)2x2ζ(3)γ3x2+3ζ(2)x+3γ2x+6x)\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

112(2γ3x2γ(π2x2+12)+x(π24xζ(3))+6γ2x+12x)\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]

2ε+2mε+12εmψ(1)(m)εψ(1)(m)2+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]

14εζ(2)m+2mε+14εmψ(0)(m)2+34εmψ(1)(m)12εmS11(m1)\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)

2ε2ζ(2)+2ε3  Li3(1x)+2ε2  Li2(1x)4ε3  Li2(1x)log(x)4ε3S12(1x)2ε3ζ(2)log(1x)+4ε3ζ(2)log(x)16ε3log3(1x)2ε3log(1x)log2(x)+ε3log2(1x)log(x)12ε2log2(1x)+2ε2log(1x)log(x)εlog(1x)+2ε3ζ(3)+1-2 \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 2F1{}_2 F_1 tabulated in Series2.m. The interested user can consult the source code (search for HYPERLIST).