Hyperbolic sine integral function
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Shi(x) represents the hyperbolic sine integral .
x is a floating-point number, then
floating-point results. The special values
Shi(0) = 0,
Shi(- ∞) = -∞ are
implemented. For all other arguments,
symbolic function calls.
x is a negative integer or a negative
rational number, then
Shi(x) = -Shi(-x). The
also uses this reflection rule when the argument is a symbolic product
involving such a factor. See Example 2.
When called with a floating-point 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 floating-point numbers, use
Alternatively, use a floating-point 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,
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(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)
i*Si(x) = Shi(i*x) for all
the complex plane.
Reference: M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions”, Dover Publications Inc., New York (1965).