# wrightOmega

Wright omega function

## Syntax

``wrightOmega(x)``

## Description

example

````wrightOmega(x)` computes the Wright omega function of `x`. If `z` is a matrix, `wrightOmega` acts elementwise on `z`.```

## Examples

### Compute Wright Omega Function of Numeric Inputs

Compute the Wright omega function for these numbers. Because these numbers are not symbolic objects, you get floating-point results:

`wrightOmega(1/2)`
```ans = 0.7662```
`wrightOmega(pi)`
```ans = 2.3061wrightOmega(-1+i*pi)```
```ans = -1.0000 + 0.0000```

### Compute Wright Omega Function of Symbolic Numbers

Compute the Wright omega function for the numbers converted to symbolic objects. For most symbolic (exact) numbers, `wrightOmega` returns unresolved symbolic calls:

`wrightOmega(sym(1/2))`
```ans = wrightOmega(1/2)```
`wrightOmega(sym(pi))`
```ans = wrightOmega(pi)```

For some exact numbers, `wrightOmega` has special values:

`wrightOmega(-1+i*sym(pi))`
```ans = -1```

### Compute Wright Omega Function of Symbolic Expression

Compute the Wright omega function for `x` and `sin(x) + x*exp(x)`. For symbolic variables and expressions, `wrightOmega` returns unresolved symbolic calls:

```syms x wrightOmega(x) wrightOmega(sin(x) + x*exp(x))```
```ans = wrightOmega(x) ans = wrightOmega(sin(x) + x*exp(x))```

### Compute Derivative of Wright Omega Function

Now compute the derivatives of these expressions:

```diff(wrightOmega(x), x, 2) diff(wrightOmega(sin(x) + x*exp(x)), x)```
```ans = wrightOmega(x)/(wrightOmega(x) + 1)^2 -... wrightOmega(x)^2/(wrightOmega(x) + 1)^3 ans = (wrightOmega(sin(x) + x*exp(x))*(cos(x) +... exp(x) + x*exp(x)))/(wrightOmega(sin(x) + x*exp(x)) + 1)```

### Compute Wright Omega Function for Matrix Input

Compute the Wright omega function for elements of matrix `M` and vector `V`:

```M = [0 pi; 1/3 -pi]; V = sym([0; -1+i*pi]); wrightOmega(M) wrightOmega(V)```
```ans = 0.5671 2.3061 0.6959 0.0415 ans = lambertw(0, 1) -1```

## Input Arguments

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Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.

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### Wright omega Function

The Wright omega function is defined in terms of the Lambert W function:

`$\omega \left(x\right)={W}_{⌈\frac{\mathrm{Im}\left(x\right)-\pi }{2\pi }⌉}\left({e}^{x}\right)$`

The Wright omega function ω(x) is a solution of the equation Y + log(Y) = X.

## References

[1] Corless, R. M. and D. J. Jeffrey. “The Wright omega Function.” Artificial Intelligence, Automated Reasoning, and Symbolic Computation (J. Calmet, B. Benhamou, O. Caprotti, L. Henocque, and V. Sorge, eds.). Berlin: Springer-Verlag, 2002, pp. 76-89.

## Version History

Introduced in R2011b