Formula:960

Digital Library of Mathematical Functions ID 2.10.E2

${\displaystyle{\displaystyle\sum_{j=a}^{n}f(j)=\int_{a}^{n}f(x)\mathrm{d}x+\tfrac{1}{2}f(a)+\tfrac{1}{2}f(n)-2\int_{0}^{\infty}\frac{\Im(f(a+iy))}{e^{2\pi y}-1}\mathrm{d}y+\sum_{s=1}^{m}\frac{B_{2s}}{(2s)!}f^{(2s-1)}(n)+2\frac{(-1)^{m}}{(2m)!}\int_{0}^{\infty}\Im(f^{(2m)}(n+i\vartheta_{n}y))\frac{y^{2m}\mathrm{d}y}{e^{2\pi y}-1},}}$

Symbols List

• ${\displaystyle{\displaystyle B_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\Im}}$: imaginary part
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle a}}$: integer
• ${\displaystyle{\displaystyle m}}$: integer
• ${\displaystyle{\displaystyle n}}$: integer
• ${\displaystyle{\displaystyle f(x)}}$: integrable function
 

Resources that cite this formula

No records found.