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# 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 the Sine Integral Function

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

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

### Handle Expressions Containing the 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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### X — Inputsymbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix

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] Cautschi, 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.