1. **Stating the problem:** Lerch's Theorem is a result in complex analysis and number theory that relates the Lerch transcendent function to other special functions. 2. **Definition:** The Lerch transcendent function is defined as $$\Phi(z,s,a) = \sum_{n=0}^\infty \frac{z^n}{(n+a)^s}$$ where $|z|<1$, $a \neq 0,-1,-2,\ldots$, and $s$ is a complex parameter. 3. **Lerch's Theorem:** It states that the Lerch transcendent can be analytically continued beyond its initial domain and relates to the Hurwitz zeta function and polylogarithm functions. Specifically, for $z=1$, $$\Phi(1,s,a) = \zeta(s,a)$$ where $\zeta(s,a)$ is the Hurwitz zeta function. 4. **Important rules:** The series converges for $|z|<1$ and can be extended analytically elsewhere. The function generalizes many special functions, including the polylogarithm $\mathrm{Li}_s(z)$ and the Hurwitz zeta function. 5. **Intermediate work:** For example, when $a=1$, $$\Phi(z,s,1) = \mathrm{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s}$$ which is the polylogarithm. 6. **Explanation:** Lerch's Theorem provides a framework to understand and extend these special functions, showing their interrelations and analytic continuations. **Final answer:** Lerch's Theorem connects the Lerch transcendent function to the Hurwitz zeta and polylogarithm functions, allowing analytic continuation and unifying these special functions under one framework.