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Hyperbolic sine integral function

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Shi(x) represents the hyperbolic sine integral 0xsinh(t)tdt.

If x is a floating-point number, then Shi(x) returns floating-point 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.

Environment Interactions

When called with a floating-point argument, the functions are sensitive to the environment variable DIGITS which determines the numerical working precision.


Example 1

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 float:

float(Shi(1)), float(Shi(sqrt(2)))

Alternatively, use a floating-point value as an argument:

Shi(-5.0), Shi(1.0), Shi(2.0 + 10.0*I)

Example 2

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)

Example 3

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)



An arithmetical expression

Return Values

Arithmetical expression.

Overloaded By



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).

See Also

MuPAD Functions

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