# §24.7 Integral Representations

## §24.7(i) Bernoulli and Euler Numbers

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

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

## §24.7(ii) Bernoulli and Euler Polynomials

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

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

### Mellin–Barnes Integral

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