アベル・プラナの和公式の誤差項

$$\begin{gathered} i{\int }_0^{\mathrm{\infty }}\frac{f(a+iy)}{e^{2\pi y}{e}^{-2\pi ia}-1}-\frac{f(a-iy)}{e^{2\pi y}{e}^{2\pi ia}-1}dy \\ =i\sum _{k=0}^M\frac{f^{(k)}(a)}{k!}{\int }_0^{\mathrm{\infty }}(\frac{(i{)}^k{y}^k}{e^{2\pi y}{e}^{-2\pi ia}-1}-\frac{(-i{)}^k{y}^k}{e^{2\pi y}{e}^{2\pi ia}-1})dy \\ =i\sum _{k=0}^M\frac{f^{(k)}(a)}{k!}{\int }_0^{\mathrm{\infty }}(\frac{(i{)}^k{y}^k{e}^{-2\pi y}{e}^{2\pi ia}}{1-e^{-2\pi y}{e}^{2\pi ia}}-\frac{(-i{)}^k{y}^k{e}^{-2\pi y}{e}^{-2\pi ia}}{1-e^{-2\pi y}{e}^{-2\pi ia}})dy \\ =i\sum _{k=0}^M\frac{f^{(k)}(a)}{k!}{\int }_0^{\mathrm{\infty }}(\sum _{n=1}^{\mathrm{\infty }}(i{)}^k{y}^k{e}^{-2\pi ny}{e}^{2\pi ian}-(-i{)}^k{y}^k{e}^{-2\pi ny}{e}^{-2\pi ian})dy(2\pi ny=t) \\ =i\sum _{k=0}^M\frac{f^{(k)}(a)}{k!}{\int }_0^{\mathrm{\infty }}(\sum _{n=1}^{\mathrm{\infty }}\frac{e^{2\pi ian}}{(2\pi n{)}^{k+1}}(i{)}^k{t}^k{e}^{-t}-\frac{e^{-2\pi ian}}{(2\pi n{)}^{k+1}}(-i{)}^k{t}^k{e}^{-t})dt \\ =i\sum _{k=0}^M\frac{f^{(k)}(a)}{k!}(\sum _{n=1}^{\mathrm{\infty }}\frac{e^{2\pi ian}}{(2\pi n{)}^{k+1}}(i{)}^{k}k!-\frac{e^{-2\pi ian}}{(2\pi n{)}^{k+1}}(-i{)}^{k}k!) \\ =\sum _{k=0}^M\frac{f^{(k)}(a)}{k!}(\sum _{n=1}^{\mathrm{\infty }}\frac{e^{2\pi ian+\frac{2\pi i(k+1)}{4}}}{(2\pi n{)}^{k+1}}k!+\frac{e^{-2\pi ian-\frac{2\pi i(k+1)}{4}}}{(2\pi n{)}^{k+1}}k!) \\ =\sum _{k=0}^M\frac{f^{(k)}(a)}{k!}\sum _{n=1}^{\mathrm{\infty }}\frac{2\cos \left(2\pi an+\frac{2\pi (k+1)}{4}\right)}{(2\pi n{)}^{k+1}}k! \\ =\sum _{k=0}^M\frac{f^{(k)}(a)}{(k+1)!}\sum _{n=1}^{\mathrm{\infty }}\frac{2\cos \left(2\pi (-a)n-\frac{2\pi (k+1)}{4}\right)}{(2\pi n{)}^{k+1}}(k+1)! \\ =-\sum _{k=0}^M\frac{f^{(k)}(a)}{(k+1)!}{\hat{B}}_{k+1}(-a)(0<a<1なら) \\ =-\sum _{k=0}^M\frac{f^{(k)}(a)}{(k+1)!}{B}_{k+1}(1-a) \\ =\sum _{k=0}^M\frac{(-1{)}^k{f}^{(k)}(a)}{(k+1)!}{B}_{k+1}(a) \end{gathered}$$
以上より

$$\begin{gathered} i{\int }_0^{\mathrm{\infty }}\frac{f(a+iy)}{e^{2\pi y}{e}^{-2\pi ia}-1}-\frac{f(a-iy)}{e^{2\pi y}{e}^{2\pi ia}-1}-\frac{f(b+iy)}{e^{2\pi y}{e}^{-2\pi ib}-1}+\frac{f(b-iy)}{e^{2\pi y}{e}^{2\pi ib}-1}dy \\ =-\sum _{k=0}^M\frac{1}{(k+1)!}(f^{(k)}(a){\hat{B}}_{k+1}(-a)-f^{(k)}(b){\hat{B}}_{k+1}(-b)) \end{gathered}$$

補足
$M$は多項式$f(x)$の次数
${\hat{B}}_k(x)$は周期ベルヌーイ多項式