1.62.1 The Laplace transform of sinhax\sinh ax is
Indeed, we have ∫0∞eaxe-sxdx=1s-a\int_{0}^{\infty}e^{ax}e^{-sx}\,dx={\frac{1}{s-a}} and ∫0∞e-axe-sxdx=1s+a\int_{0}^{\infty}e^{-ax}e^{-sx}\,dx={\frac{1}{s+a}} by our assumption on aa, hence H(s)=12(1s-a-1s+a)=12s+a-(s-a)(s-a)(s+a).H(s)={\frac{1}{2}(\frac{1}{s-a}-\frac{1}{s+a})}\,{=\frac{1}{2}\frac{s+a-(s-a)}% {(s-a)(s+a)}.}
1.62.2 Question: What is the Laplace transform of coshax\cosh ax?
Remark. The Laplace transforms of sin,cos,sinh\sin,\cos,\sinh and cosh\cosh are all rational functions.