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(2)
where is the hyperbolic cosine. It is implemented in Mathematica as Sech[z].
On the real line, it has a maximum at x = 0 and inflection points at (Sloane's A091648). It
has a fixed point at x = 0,76500995… (Sloane's A069814).
The derivative is given by
, (3)
where is the hyperbolic tangent, and the indefinite integral by
where is a constant of integration.
has the Taylor series
(Sloane's A046976 and A046977), where En is an Euler number and n! is a factorial.
Equating coefficients of , , and in the Ramanujan cos/cosh identity
gives the amazing identities
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(4)
(5)
(6)
(7)
(8)