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wrightOmega

Wright omega function

Syntax

```wrightOmega(x) wrightOmega(A) ```

Description

`wrightOmega(x)` computes the Wright omega function of `x`.

`wrightOmega(A)` computes the Wright omega function of each element of `A`.

Input Arguments

 `x` Number, symbolic variable, or symbolic expression. `A` Vector or matrix of numbers, symbolic variables, or symbolic expressions.

Examples

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

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 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))```

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 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```

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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

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.