$\int_0^1\:Li_s(x)\:dx$ integralini çözün

Question

$Li_s(x)$ polylogaritma fonksiyonu olmak üzere :

$$\int_0^1\:Li_s(x)\:dx$$

İntegralini çözün.

Answer 1

İntegralimiz :

$$\int_0^1\:Li_s(x)\:dx$$

$Li_s(x)$ ifadesini açalım.

$$\int_0^1\:\sum_{k=0}^\infty\:\frac{x^k}{k^p}\:dx$$

$$\sum_{k=0}^\infty\frac{1}{k^p}\:\int_0^1\:x^k\:dx$$

İntegarali çözelim.

$$\sum_{k=0}^\infty\frac{1}{k^p}\:\frac{x^{k+1}}{k+1}\Bigg|_0^1$$

$$\sum_{k=0}^\infty\frac{1}{k^p(k+1)}$$

Sadeleştirelim.

$$\sum_{k=0}^\infty\Bigg(\frac{1}{k^p}-\frac{1}{k^{p-1}(k+1)}\Bigg)$$

$$\sum_{k=0}^\infty\Bigg(\frac{1}{k^p}-\frac{1}{k^{p-1}}+\frac{1}{k^{p-2}(1+k)}\Bigg)$$

$$\sum_{k=0}^\infty\Bigg(\frac{1}{k^p}-\frac{1}{k^{p-1}}+\frac{1}{k^{p-2}}-\cdots(-1)^{p+1}\frac{1}{k(k+1)}\Bigg)$$

$$\zeta(p)-\zeta(p-1)+\zeta(p-2)-\cdots(-1)^{p+1}$$

İfadeyi toplama sembolü ile yazalım.

$$\large\color{#A00000}{\boxed{\int_0^1\:Li_s(x)\:dx=(-1)^{p+1}+\sum_{k=0}^{p-1}\:(-1)^k\,\zeta(p-k)}}$$