Hyperbolic sine integral function
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Shi(x
)
Shi(x)
represents the hyperbolic sine integral $$\underset{0}{\overset{x}{\int}}\frac{\mathrm{sinh}\left(t\right)}{t}dt$$.
If x
is a floatingpoint number, then Shi(x)
returns
floatingpoint results. The special values Shi(0) = 0
, Shi(∞)
= ∞
, Shi( ∞) = ∞
are
implemented. For all other arguments, Shi
returns
symbolic function calls.
If x
is a negative integer or a negative
rational number, then Shi(x) = Shi(x)
. The Shi
function
also uses this reflection rule when the argument is a symbolic product
involving such a factor. See Example 2.
When called with a floatingpoint argument, the functions are
sensitive to the environment variable DIGITS
which determines
the numerical working precision.
Most calls with exact arguments return themselves unevaluated:
Shi(0), Shi(1), Shi(sqrt(2)), Shi(x + 1), Shi(infinity)
To approximate exact results with floatingpoint numbers, use float
:
float(Shi(1)), float(Shi(sqrt(2)))
Alternatively, use a floatingpoint value as an argument:
Shi(5.0), Shi(1.0), Shi(2.0 + 10.0*I)
For negative real numbers and products involving such numbers, Shi
applies
the reflection rule Shi(x) = Shi(x)
:
Shi(3), Shi(3/7), Shi(sqrt(2)), Shi(x/7), Shi(0.3*x)
No such "normalization" occurs for complex numbers or arguments that are not products:
Shi( 3  I), Shi(3 + I), Shi(x  1), Shi(1  x)
diff
, float
, limit
, series
, and other functions
handle expressions involving Shi
:
diff(Shi(x), x, x, x), float(ln(3 + Shi(sqrt(PI))))
limit(Shi(2*I*x^2/(1+x)), x = infinity)
series(Shi(x), x = 0)
series(Shi(I*x), x = infinity, 3)

Arithmetical expression.
x
Si
, Ssi
, and Shi
are
entire functions.
i*Si(x) = Shi(i*x)
for all x
in
the complex plane.
Reference: M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions", Dover Publications Inc., New York (1965).