coshz=ez+e−z2{\displaystyle \cosh z={\frac {e^{z}+e^{-z}}{2}}}
sinhz=ez−e−z2{\displaystyle \sinh z={\frac {e^{z}-e^{-z}}{2}}}
coshiz=cosz{\displaystyle \cosh iz=\cos z}
sinhiz=isinz{\displaystyle \sinh iz=i\sin z}
tanhiz=itanz{\displaystyle \tanh iz=i\tan z}
cosh(x+y)=coshxcoshy+sinhxsinhy{\displaystyle \cosh(x+y)=\cosh x\cosh y+\sinh x\sinh y}
sinh(x+y)=sinhxcoshy+coshxsinhy{\displaystyle \sinh(x+y)=\sinh x\cosh y+\cosh x\sinh y}
cosh(x−y)=coshxcoshy−sinhxsinhy{\displaystyle \cosh(x-y)=\cosh x\cosh y-\sinh x\sinh y}
sinh(x−y)=sinhxcoshy−coshxsinhy{\displaystyle \sinh(x-y)=\sinh x\cosh y-\cosh x\sinh y}