アベル・プラナの和公式
∫ 0 ∞ y 2 k − 1 e 2 π y − 1 d y {\displaystyle {\begin{aligned}\sum _{n=a}^{b}f(n)&={\frac {1}{2}}f(a)+{\frac {1}{2}}f(b)+\int _{a}^{b}f(z)dz+i\int _{0}^{\infty }{\frac {f(a+iy)-f(a-iy)-f(b+iy)+f(b-iy)}{e^{2{\pi }y}-1}}dy\\&={\frac {1}{2}}f(a)+{\frac {1}{2}}f(b)+\int _{a}^{b}f(z)dz+i\int _{0}^{\infty }\sum _{k=0}^{\infty }{\frac {f^{(k)}(a)(iy)^{k}-f^{(k)}(a)(-iy)^{k}-f^{(k)}(b)(iy)^{k}+f^{(k)}(b)(-iy)^{k}}{\left(e^{2{\pi }y}-1\right)k!}}dy\\&={\frac {1}{2}}f(a)+{\frac {1}{2}}f(b)+\int _{a}^{b}f(z)dz+2\sum _{k=1}^{\infty }(-1)^{k}{\frac {f^{(2k-1)}(a)-f^{(2k-1)}(b)}{(2k-1)!}}\int _{0}^{\infty }{\frac {y^{2k-1}}{e^{2{\pi }y}-1}}dy\\\end{aligned}}}