指数積分
d t = e − z / 2 ∑ n = 1 ∞ ( ∑ k = 0 ⌊ ( n − 1 ) / 2 ⌋ 2 2 k + 1 ) ( z / 2 ) n n ! {\displaystyle {\begin{aligned}\operatorname {Ein} (z)&=\int _{0}^{z}e^{-t}\sum _{k=1}^{\infty }{\frac {t^{k-1}}{k!}}\,\operatorname {d} \!t\\&=e^{-z}\sum _{n=1}^{\infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}\right){\frac {z^{n}}{n!}}\\\operatorname {Ein} (z)&=\int _{0}^{z}e^{-t/2}\sum _{k=0}^{\infty }{\frac {(t/2)^{2k}}{(2k+1)!}}\,\operatorname {d} \!t\\&=e^{-z/2}\sum _{n=1}^{\infty }\left(\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{\frac {2}{2k+1}}\right){\frac {(z/2)^{n}}{n!}}\end{aligned}}}