Hyperbolic Functions

f$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ F$\left(\vphantom{ s}\right.$s$\left.\vphantom{ s}\right)$ = L$\left\{\vphantom{ f}\right.$f$\left.\vphantom{ f}\right\}$$\left(\vphantom{ s}\right.$s$\left.\vphantom{ s}\right)$
sinh kt ${\dfrac{{k}}{{s^{2}-k^{2}}}}$
cosh kt ${\dfrac{{s}}{{s^{2}-k^{2}}}}$
sinh2kt ${\dfrac{{2k^{2}}}{{s\left( s^{2}-4k^{2}\right) }}}$
cosh2kt     ${\dfrac{{s^{2}-2k^{2}}}{{s\left( s^{2}-4k^{2}\right) }}}$
                
f$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ F$\left(\vphantom{ s}\right.$s$\left.\vphantom{ s}\right)$ = L$\left\{\vphantom{ f}\right.$f$\left.\vphantom{ f}\right\}$$\left(\vphantom{ s}\right.$s$\left.\vphantom{ s}\right)$
t sinh kt ${\dfrac{{2ks}}{{\left( s^{2}-k^{2}\right) ^{2}}}}$
t cosh kt ${\dfrac{{s^{2}+k^{2}}}{{\left( s^{2}-k^{2}\right) ^{2}}}}$
${\dfrac{{2\left( 1-\cosh kt\right) }}{{t}}}$     ln${\dfrac{{s^{2}-k^{2}}}{{s^{2}% }}}$