Chapter 15.1, Problem 8E

To determine

To prove: That $L^{-1}\left\{\frac{\sinh a\sqrt{s}}{s\sinh \sqrt{s}}\right\}={\displaystyle \sum _{n=0}^{\infty }\left[erf\left(\frac{2n+1+a}{2\sqrt{t}}\right)-erf\left(\frac{2n+1-a}{2\sqrt{t}}\right)\right]}$.