Shi

Hyperbolic sine integral function

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

Shi(x)

Description

Shi(x) represents the hyperbolic sine integral .

If x is a floating-point number, then Shi(x) returns numerical values. The special values Shi(0) = 0, Shi(∞) = ∞, Shi(- ∞) = - ∞ are implemented. For all other arguments, Shi returns symbolic function calls.

The reflection rule Shi(x) = - Shi(- x) is used if the argument is a negative integer or a negative rational number. It is also used if the argument is a symbolic product involving such a factor. Cf. 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.

Examples

Example 1

We demonstrate some calls with exact and symbolic input data:

Si(0), Si(1), Si(sqrt(2)), Si(x + 1), Si(infinity)

Ssi(0), Ssi(1), Ssi(sqrt(2)), Ssi(x + 1), Ssi(infinity)

Shi(0), Shi(1), Shi(sqrt(2)), Shi(x + 1), Shi(infinity)

Floating point values are computed for floating-point arguments:

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

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

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

Example 2

The reflection rule Si(- x) = - Si(x), Ssi(- x) = - Ssi(x) - π, Shi(- x) = - Shi(x) is implemented for negative real numbers and products involving such numbers:

Si(-3), Si(-3/7), Si(-sqrt(2)), Si(-x/7), Si(-0.3*x)

Ssi(-3), Ssi(-3/7), Ssi(-sqrt(2)), Ssi(-x/7), Ssi(-0.3*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:

Si(- 3 - I), Si(3 + I), Si(x - 1), Si(1 - x)

Ssi(- 3 - I), Ssi(3 + I), Ssi(x - 1), Ssi(1 - x)

Shi(- 3 - I), Shi(3 + I), Shi(x - 1), Shi(1 - x)

Example 3

The functions diff, float, limit, and series handle expressions involving Si and Shi:

diff(Si(x), x, x, x), float(ln(3 + Si(sqrt(PI))))

diff(Ssi(x), x, x, x), float(ln(3 + Ssi(sqrt(PI))))

diff(Shi(x), x, x, x), float(ln(3 + Shi(sqrt(PI))))

limit(Si(2*x^2/(1+x)), x = infinity)

limit(Ssi(2*x^2/(1+x)), x = infinity)

limit(Shi(2*I*x^2/(1+x)), x = infinity)

series(Si(x), x = 0)

series(Ssi(x), x = 0)

series(Shi(x), x = 0)

series(Si(x), x = infinity, 3)

series(Ssi(x), x = infinity, 3)

series(Shi(I*x), x = infinity, 3)

Parameters

x

An arithmetical expression

Return Values

Arithmetical expression.

Overloaded By

x

Algorithms

Si, Ssi, and Shi are entire functions.

Si and Ssi are related by Ssi(x) = Si(x) - π for all x in the complex plane.

Si and Shi are related by 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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