# §24.7 Integral Representations

## §24.7(i) Bernoulli and Euler Numbers

The identities in this subsection hold for $n=1,2,\ldots$. (24.7.6) also holds for $n=0$.

 24.7.1 $\displaystyle B_{2n}$ $\displaystyle=(-1)^{n+1}\frac{4n}{1-2^{1-2n}}\int_{0}^{\infty}\frac{t^{2n-1}}{% e^{2\pi t}+1}\mathrm{d}t=(-1)^{n+1}\frac{2n}{1-2^{1-2n}}\int_{0}^{\infty}t^{2n% -1}e^{-\pi t}\mathop{\mathrm{sech}\/}\nolimits\!\left(\pi t\right)\mathrm{d}t,$ 24.7.2 $\displaystyle B_{2n}$ $\displaystyle=(-1)^{n+1}4n\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2\pi t}-1}% \mathrm{d}t=(-1)^{n+1}2n\int_{0}^{\infty}t^{2n-1}e^{-\pi t}\mathop{\mathrm{% csch}\/}\nolimits\!\left(\pi t\right)\mathrm{d}t,$ 24.7.3 $\displaystyle B_{2n}$ $\displaystyle=(-1)^{n+1}\frac{\pi}{1-2^{1-2n}}\int_{0}^{\infty}t^{2n}{\mathop{% \mathrm{sech}\/}\nolimits^{2}}\!\left(\pi t\right)\mathrm{d}t,$ 24.7.4 $\displaystyle B_{2n}$ $\displaystyle=(-1)^{n+1}\pi\int_{0}^{\infty}t^{2n}{\mathop{\mathrm{csch}\/}% \nolimits^{2}}\!\left(\pi t\right)\mathrm{d}t,$ 24.7.5 $\displaystyle B_{2n}$ $\displaystyle=(-1)^{n}\frac{2n(2n-1)}{\pi}\*\int_{0}^{\infty}t^{2n-2}\mathop{% \ln\/}\nolimits\!\left(1-e^{-2\pi t}\right)\mathrm{d}t.$ 24.7.6 $\displaystyle E_{2n}$ $\displaystyle=(-1)^{n}2^{2n+1}\int_{0}^{\infty}t^{2n}\mathop{\mathrm{sech}\/}% \nolimits\!\left(\pi t\right)\mathrm{d}t.$

## §24.7(ii) Bernoulli and Euler Polynomials

The following four equations hold for $0<\Re{x}<1$.

 24.7.7 $\displaystyle\mathop{B_{2n}\/}\nolimits\!\left(x\right)$ $\displaystyle=(-1)^{n+1}2n\*\int_{0}^{\infty}\frac{\mathop{\cos\/}\nolimits\!% \left(2\pi x\right)-e^{-2\pi t}}{\mathop{\cosh\/}\nolimits\!\left(2\pi t\right% )-\mathop{\cos\/}\nolimits\!\left(2\pi x\right)}t^{2n-1}\mathrm{d}t,$ $n=1,2,\dots$, 24.7.8 $\displaystyle\mathop{B_{2n+1}\/}\nolimits\!\left(x\right)$ $\displaystyle=(-1)^{n+1}(2n+1)\*\int_{0}^{\infty}\frac{\mathop{\sin\/}% \nolimits\!\left(2\pi x\right)}{\mathop{\cosh\/}\nolimits\!\left(2\pi t\right)% -\mathop{\cos\/}\nolimits\!\left(2\pi x\right)}t^{2n}\mathrm{d}t.$ 24.7.9 $\displaystyle\mathop{E_{2n}\/}\nolimits\!\left(x\right)$ $\displaystyle=(-1)^{n}4\int_{0}^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(% \pi x\right)\mathop{\cosh\/}\nolimits\!\left(\pi t\right)}{\mathop{\cosh\/}% \nolimits\!\left(2\pi t\right)-\mathop{\cos\/}\nolimits\!\left(2\pi x\right)}t% ^{2n}\mathrm{d}t,$ 24.7.10 $\displaystyle\mathop{E_{2n+1}\/}\nolimits\!\left(x\right)$ $\displaystyle=(-1)^{n+1}4\*\int_{0}^{\infty}\frac{\mathop{\cos\/}\nolimits\!% \left(\pi x\right)\mathop{\sinh\/}\nolimits\!\left(\pi t\right)}{\mathop{\cosh% \/}\nolimits\!\left(2\pi t\right)-\mathop{\cos\/}\nolimits\!\left(2\pi x\right% )}t^{2n+1}\mathrm{d}t.$

### Mellin–Barnes Integral

 24.7.11 $\mathop{B_{n}\/}\nolimits\!\left(x\right)=\frac{1}{2\pi i}\int_{-c-i\infty}^{-% c+i\infty}(x+t)^{n}\left(\frac{\pi}{\mathop{\sin\/}\nolimits\!\left(\pi t% \right)}\right)^{2}\mathrm{d}t,$ $0.

## §24.7(iii) Compendia

For further integral representations see Prudnikov et al. (1986a, §§2.3–2.6) and Gradshteyn and Ryzhik (2000, Chapters 3 and 4).