Hyperbolic sine modulus identity
= Hyperbolic sine modulus identity
{title2=$|\sinh(x+iy)|^2=\sinh^2x+\sin^2y$}
For real $x,y$, the <exponential function> definitions give $\sinh(x+iy)=\sinh x\cos y+i\cosh x\sin y$. Taking the squared <complex modulus> and using $\cosh^2x-\sinh^2x=1$ gives the identity. In particular $|\sinh(x+iy)|\geq|\sinh x|$, uniformly in $y$. This controls the vertical sides of rectangular <contours> in <residue theorem> arguments with <hyperbolic functions>.