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wrightOmega

Wright omega function

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.

 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.

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