# ssinint

Shifted sine integral function

## Description

example

ssinint(X) returns the shifted sine integral function ssinint(X) = sinint(X) — pi/2.

## Examples

### Shifted Sine Integral Function for Numeric and Symbolic Arguments

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

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

A = ssinint([- pi, 0, pi/2, pi, 1])
A =
-3.4227   -1.5708   -0.2000    0.2811   -0.6247

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

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

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

vpa(symA)
ans =
[ -3.4227333787773627895923750617977,...
-1.5707963267948966192313216916398,...
-0.20003415864040813916164340325818,...
0.28114072518756955112973167851824,...
-0.62471325642771360428996837781657]

### Plot Shifted Sine Integral Function

Plot the shifted sine integral function on the interval from -4*pi to 4*pi.

syms x
fplot(ssinint(x),[-4*pi 4*pi])
grid on

### Handle Expressions Containing Shifted Sine Integral Function

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

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

syms x
diff(ssinint(x), x)
diff(ssinint(x), x, x)
ans =
sin(x)/x

ans =
cos(x)/x - sin(x)/x^2

Find the indefinite integral of the shifted sine integral function:

int(ssinint(x), x)
ans =
cos(x) + x*ssinint(x)

Find the Taylor series expansion of ssinint(x):

taylor(ssinint(x), x)
ans =
x^5/600 - x^3/18 + x - pi/2

## Input Arguments

collapse all

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

collapse all

### Sine Integral Function

The sine integral function is defined as follows:

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

### Shifted Sine Integral Function

The sine integral function is defined as Ssi(x) = Si(x) - π/2.

## 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.

## Version History

Introduced in R2014a