Relationship between dilogarithm and log(1-z)/z

Theorem

The following formula holds: $$\mathrm{Li}_2(z)=-\displaystyle\int_0^z \dfrac{\log(1-z)}{z} \mathrm{d}z,$$ where $\mathrm{Li}_2$ denotes the dilogarithm and $\log$ denotes the logarithm.

References

• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.11.1 (22)$
• 1958: Leonard Lewin: Dilogarithms and Associated Functions ... (previous) ... (next): $(1.3)$
• 1981: Leonard Lewin: Polylogarithms and Associated Functions (2nd ed.) ... (previous) ... (next): $(1.3)$
• 1991: Leonard Lewin: Structural Properties of Polylogarithms ... (previous) ... (next): $(1.2)$