/ / 2\
| | 1 + x |
|lerchphi|x, 1, ------|
| | 2 | / / / 2 2 \ / 2 2 \ 2 2 \ / / 2 2 \ / 2 2 \ 2 2 \ / / 2 2 \ / 2 2 \ 2 2 / 2 2 \ / 2 2 \ 2 2 \\
| \ x / | | | im (x) re (x) | / 2 2 \ | re (x) im (x) | / 2 2 \ 4*im (x)*re (x) | | | im (x) re (x) | / 2 2 \ | re (x) im (x) | / 2 2 \ 4*im (x)*re (x) | | | im (x) re (x) | / 2 2 \ | re (x) im (x) | / 2 2 \ 4*im (x)*re (x) | im (x) re (x) | / 2 2 \ | re (x) im (x) | / 2 2 \ 4*im (x)*re (x) ||
|---------------------- for Or|And||x| <= 1, 1 + |------------------ - ------------------|*\1 - 2*im (x) + 2*re (x)/ + |------------------ - ------------------|*\1 + re (x) - im (x)/ - ------------------ < 0|, And|1 + |------------------ - ------------------|*\1 - 2*im (x) + 2*re (x)/ + |------------------ - ------------------|*\1 + re (x) - im (x)/ - ------------------ >= 1, |x| < 1|, And|1 + |------------------ - ------------------|*\1 - 2*im (x) + 2*re (x)/ + |------------------ - ------------------|*\1 + re (x) - im (x)/ - ------------------ >= 0, |x| <= 1, 1 + |------------------ - ------------------|*\1 - 2*im (x) + 2*re (x)/ + |------------------ - ------------------|*\1 + re (x) - im (x)/ - ------------------ < 1, x != 1||
| x | | | 2 2| | 2 2| 2 | | | 2 2| | 2 2| 2 | | | 2 2| | 2 2| 2 | 2 2| | 2 2| 2 ||
| | | |/ 2 2 \ / 2 2 \ | |/ 2 2 \ / 2 2 \ | / 2 2 \ | | |/ 2 2 \ / 2 2 \ | |/ 2 2 \ / 2 2 \ | / 2 2 \ | | |/ 2 2 \ / 2 2 \ | |/ 2 2 \ / 2 2 \ | / 2 2 \ |/ 2 2 \ / 2 2 \ | |/ 2 2 \ / 2 2 \ | / 2 2 \ ||
| \ \ \\im (x) + re (x)/ \im (x) + re (x)/ / \\im (x) + re (x)/ \im (x) + re (x)/ / \im (x) + re (x)/ / \ \\im (x) + re (x)/ \im (x) + re (x)/ / \\im (x) + re (x)/ \im (x) + re (x)/ / \im (x) + re (x)/ / \ \\im (x) + re (x)/ \im (x) + re (x)/ / \\im (x) + re (x)/ \im (x) + re (x)/ / \im (x) + re (x)/ \\im (x) + re (x)/ \im (x) + re (x)/ / \\im (x) + re (x)/ \im (x) + re (x)/ / \im (x) + re (x)/ //
|
< oo
| ____
| \ `
| \ n
| \ x
| ) -------- otherwise
| / 2
| / 1 + n*x
| /___,
| n = 1
\
Piecewise((lerchphi(x, 1, (1 + x^2)/x^2)/x, ((|x| <= 1)∧(1 + (im(x)^2/(im(x)^2 + re(x)^2)^2 - re(x)^2/(im(x)^2 + re(x)^2)^2)*(1 - 2*im(x)^2 + 2*re(x)^2) + (re(x)^2/(im(x)^2 + re(x)^2)^2 - im(x)^2/(im(x)^2 + re(x)^2)^2)*(1 + re(x)^2 - im(x)^2) - 4*im(x)^2*re(x)^2/(im(x)^2 + re(x)^2)^2 < 0))∨((|x| < 1)∧(1 + (im(x)^2/(im(x)^2 + re(x)^2)^2 - re(x)^2/(im(x)^2 + re(x)^2)^2)*(1 - 2*im(x)^2 + 2*re(x)^2) + (re(x)^2/(im(x)^2 + re(x)^2)^2 - im(x)^2/(im(x)^2 + re(x)^2)^2)*(1 + re(x)^2 - im(x)^2) - 4*im(x)^2*re(x)^2/(im(x)^2 + re(x)^2)^2 >= 1))∨((Ne(x, 1))∧(|x| <= 1)∧(1 + (im(x)^2/(im(x)^2 + re(x)^2)^2 - re(x)^2/(im(x)^2 + re(x)^2)^2)*(1 - 2*im(x)^2 + 2*re(x)^2) + (re(x)^2/(im(x)^2 + re(x)^2)^2 - im(x)^2/(im(x)^2 + re(x)^2)^2)*(1 + re(x)^2 - im(x)^2) - 4*im(x)^2*re(x)^2/(im(x)^2 + re(x)^2)^2 >= 0)∧(1 + (im(x)^2/(im(x)^2 + re(x)^2)^2 - re(x)^2/(im(x)^2 + re(x)^2)^2)*(1 - 2*im(x)^2 + 2*re(x)^2) + (re(x)^2/(im(x)^2 + re(x)^2)^2 - im(x)^2/(im(x)^2 + re(x)^2)^2)*(1 + re(x)^2 - im(x)^2) - 4*im(x)^2*re(x)^2/(im(x)^2 + re(x)^2)^2 < 1))), (Sum(x^n/(1 + n*x^2), (n, 1, oo)), True))