sinhint

Hyperbolic sine integral function

Description

example

sinhint(X) returns the hyperbolic sine integral function of X.

Examples

Hyperbolic Sine Integral Function for Numeric and Symbolic Arguments

Depending on its arguments, sinhint returns floating-point or exact symbolic results.

Compute the hyperbolic sine integral function for these numbers. Because these numbers are not symbolic objects, sinhint returns floating-point results.

A = sinhint([-pi, -1, 0, pi/2, 2*pi])
A =
-5.4696   -1.0573         0    1.8027   53.7368

Compute the hyperbolic sine integral function for the numbers converted to symbolic objects. For many symbolic (exact) numbers, sinhint returns unresolved symbolic calls.

symA = sinhint(sym([-pi, -1, 0, pi/2, 2*pi]))
symA =
[ -sinhint(pi), -sinhint(1), 0, sinhint(pi/2), sinhint(2*pi)]

Use vpa to approximate symbolic results with floating-point numbers:

vpa(symA)
ans =
[ -5.4696403451153421506369580091277,...
-1.0572508753757285145718423548959,...
0,...
1.802743198288293882089794577617,...
53.736750620859153990408011863262]

Plot Hyperbolic Sine Integral Function

Plot the hyperbolic sine integral function on the interval from -2*pi to 2*pi.

syms x
fplot(sinhint(x),[-2*pi 2*pi])
grid on

Handle Expressions Containing Hyperbolic Sine Integral Function

Many functions, such as diff, int, and taylor, can handle expressions containing sinhint.

Find the first and second derivatives of the hyperbolic sine integral function:

syms x
diff(sinhint(x), x)
diff(sinhint(x), x, x)
ans =
sinh(x)/x

ans =
cosh(x)/x - sinh(x)/x^2

Find the indefinite integral of the hyperbolic sine integral function:

int(sinhint(x), x)
ans =
x*sinhint(x) - cosh(x)

Find the Taylor series expansion of sinhint(x):

taylor(sinhint(x), x)
ans =
x^5/600 + x^3/18 + x

Input Arguments

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Input, specified as a symbolic number, variable, expression, or function, or as a vector or matrix of symbolic numbers, variables, expressions, or functions.

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Hyperbolic Sine Integral Function

The hyperbolic sine integral function is defined as follows:

$\text{Shi}\left(x\right)=\underset{0}{\overset{x}{\int }}\frac{\mathrm{sinh}\left(t\right)}{t}dt$

References

[1] Gautschi, W. and W. F. Cahill. “Exponential Integral and Related Functions.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.