Sinhc Function
المؤلف:
Sloane, N. J. A
المصدر:
Sequences A133916 and A133917 in
الجزء والصفحة:
...
24-4-2020
1061
Sinhc Function
By analogy with the sinc function, define the sinhc function by
![sinhc(z)=<span style=]() {(sinhz)/z for z!=0; 1 for z=0. " src="https://mathworld.wolfram.com/images/equations/SinhcFunction/NumberedEquation1.gif" style="height:60px; width:180px" /> |
(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." https://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."
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