不定和分
∑ x tan  a x = i x − 1 a ψ e 2 i a ( x − π 2 a ) + C ( a ≠ n π 2 ) , {\displaystyle \sum _{x}\tan ax=ix-{\frac {1}{a}}\psi _{e^{2ia}}\left(x-{\frac {\pi }{2a}}\right)+C\quad (a\neq {\frac {n\pi }{2}}),} ただし、ψq(x) は q-ディガンマ函数.
∑ x tan  x = i x − ψ e 2 i ( x + π 2 ) + C = − ∑ k = 1 ∞ ( ψ ( k π − π 2 + 1 − z ) + ψ ( k π − π 2 + z ) − ψ ( k π − π 2 + 1 ) − ψ ( k π − π 2 ) ) + C , {\displaystyle {\begin{aligned}\sum _{x}\tan x&=ix-\psi _{e^{2i}}\left(x+{\frac {\pi }{2}}\right)+C\\&=-\sum _{k=1}^{\infty }\left(\psi \left(k\pi -{\frac {\pi }{2}}+1-z\right)+\psi \left(k\pi -{\frac {\pi }{2}}+z\right)-\psi \left(k\pi -{\frac {\pi }{2}}+1\right)-\psi \left(k\pi -{\frac {\pi }{2}}\right)\right)+C\end{aligned}},}
∑ i = a b f ( i ) Δ g ( i ) = f ( b + 1 ) g ( b + 1 ) − f ( a ) g ( a ) − ∑ i = a b g ( i + 1 ) Δ f ( i ) . {\displaystyle \sum _{i=a}^{b}f(i)\Delta g(i)=f(b+1)g(b+1)-f(a)g(a)-\sum _{i=a}^{b}g(i+1)\Delta f(i).}