# sinint

Sine integral function

## Description

example

sinint(X) returns the sine integral function of X.

## Examples

### Sine Integral Function for Numeric and Symbolic Arguments

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

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

A = sinint([- pi, 0, pi/2, pi, 1])
A =
-1.8519         0    1.3708    1.8519    0.9461

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

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

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

vpa(symA)
ans =
[ -1.851937051982466170361053370158,...
0,...
1.3707621681544884800696782883816,...
1.851937051982466170361053370158,...
0.94608307036718301494135331382318]

### Plot Sine Integral Function

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

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

### Handle Expressions Containing Sine Integral Function

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

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

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

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

Find the indefinite integral of the sine integral function:

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

Find the Taylor series expansion of sinint(x):

taylor(sinint(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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### Sine Integral Function

The sine integral function is defined as follows:

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

## 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 before R2006a