Sinhc Function

By analogy with the sinc function, define the sinhc function by

(1)

Since  is not a cardinal function, the "analogy" with the sinc function is one of functional structure, not mathematical properties. It is quite possible that a better term than  could be coined, although there appears to be no other name previously assigned to this function.

The function has derivative

(2)

and indefinite integral

(3)

where  is the Shi function.

The function has real fixed points at 1.31328371835... (OEIS A133916) and 2.63924951389... (OEIS A133917).

REFERENCES:

Holin, H. "C++ Boost Special Functions Library." http://www.boost.org/doc/libs/1_46_1/boost/math/special_functions/sinhc.hpp.

Sloane, N. J. A. Sequences A133916 and A133917 in "The On-Line Encyclopedia of Integer Sequences."