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]
21δlog(1−x)+21δlog(x)+1
21ζ(2)x+x1
Series[Gamma[x], {x, 0, 1}]
x1−γ+121(6γ2+π2)x+O(x2)
−3x2ζ(3)+21ζ(2)x+x1
Series2[Gamma[x], x, 2, FinalSubstitutions -> {}] // FullSimplify
61(−3γ(ζ(2)x2+2)−2x2ζ(3)−γ3x2+3ζ(2)x+3γ2x+x6)
Series[Gamma[x], {x, 0, If[$VersionNumber < 5, 4, 2]}] // Normal // Expand // FullSimplify
121(−2γ3x2−γ(π2x2+12)+x(π2−4xζ(3))+6γ2x+x12)
There is a table of expansions of special hypergeometric functions.
Series2[HypergeometricPFQ[{1, OPEm - 1, Epsilon/2 + OPEm}, {OPEm, OPEm + Epsilon}, 1], Epsilon, 1]
−ε2+ε2m+21εmψ(1)(m)−2εψ(1)(m)+1
Series2[HypergeometricPFQ[{1, OPEm, Epsilon/2 + OPEm}, {1 + OPEm, Epsilon + OPEm}, 1], Epsilon, 1]
41εζ(2)m+ε2m+41εmψ(0)(m)2+43εmψ(1)(m)−21εmS11(m−1)
Hypergeometric2F1[1, Epsilon, 1 + 2 Epsilon, x]
Series2[%, Epsilon, 3]
\, _2F_1(1,\varepsilon ;2 \varepsilon +1;x)
−2ε2ζ(2)+2ε3Li3(1−x)+2ε2Li2(1−x)−4ε3Li2(1−x)log(x)−4ε3S12(1−x)−2ε3ζ(2)log(1−x)+4ε3ζ(2)log(x)−61ε3log3(1−x)−2ε3log(1−x)log2(x)+ε3log2(1−x)log(x)−21ε2log2(1−x)+2ε2log(1−x)log(x)−εlog(1−x)+2ε3ζ(3)+1
There are over 100 more special expansions of 2F1 tabulated in Series2.m
. The interested user can consult the source code (search for HYPERLIST).