Lambert W function

The Lambert W function `W(x)`

is
a set of solutions of the equation `x = W(x)e`

.^{W(x)}

Solve this equation. The solutions is the Lambert W function.

syms x W solve(x == W*exp(W), W)

ans = lambertw(0, x)

Verify that various branches of the Lambert W function are valid
solutions of the equation `x = W*e`

:^{W}

k = -2:2 syms x isAlways(x - subs(W*exp(W), W, lambertw(k,x)) == 0)

k = -2 -1 0 1 2 ans = 1 1 1 1 1

Depending on its arguments, `lambertw`

can
return floating-point or exact symbolic results.

Compute the Lambert W functions for these numbers. Because these numbers are not symbolic objects, you get floating-point results.

A = [0 -1/exp(1); pi i]; lambertw(A) lambertw(-1, A)

ans = 0.0000 + 0.0000i -1.0000 + 0.0000i 1.0737 + 0.0000i 0.3747 + 0.5764i ans = -Inf + 0.0000i -1.0000 + 0.0000i -0.3910 - 4.6281i -1.0896 - 2.7664i

Compute the Lambert W functions for the numbers converted to
symbolic objects. For most symbolic (exact) numbers, `lambertw`

returns
unresolved symbolic calls.

A = [0 -1/exp(sym(1)); pi i]; W0 = lambertw(A) Wmin1 = lambertw(-1, A)

W0 = [ 0, -1] [ lambertw(0, pi), lambertw(0, 1i)] Wmin1 = [ -Inf, -1] [ lambertw(-1, pi), lambertw(-1, 1i)]

Use `vpa`

to approximate symbolic results
with the required number of digits:

vpa(W0, 10) vpa(Wmin1, 5)

ans = [ 0, -1.0] [ 1.073658195, 0.3746990207 + 0.576412723i] ans = [ -Inf, -1.0] [ - 0.39097 - 4.6281i, - 1.0896 - 2.7664i]

Plot the principal branch of the Lambert W function on the complex plane.

Create the combined mesh and contour plot of the real value of the Lambert W function on the complex plane.

syms x y real ezmeshc(real(lambertw(x + i*y)), [-100, 100, -100, 100])

Now, plot the imaginary value of the Lambert W function on the complex plane. This function has a branch cut along the negative real axis. For better perspective, create the mesh and contour plots separately.

ezmesh(imag(lambertw(x + i*y)), [-100, 100, -100, 100])

ezcontourf(imag(lambertw(x + i*y)), [-100, 100, -100, 100])

Plot the absolute value of the Lambert W function on the complex plane.

ezmeshc(abs(lambertw(x + i*y)), [-100, 100, -100, 100])

For further computations, clear the assumptions on `x`

and `y`

:

syms x y clear

Plot the two main branches, and , of the Lambert W function.

Plot the principal branch :

```
syms x
ezplot(lambertw(x))
```

Add the branch :

```
hold on
ezplot(lambertw(-1, x))
```

Adjust the axes limits and add the title:

```
axis([-0.5, 4, -4, 2])
title('Lambert W function, two main branches')
```

[1] Corless, R.M, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey,
and D.E. Knuth "On the Lambert W Function" *Advances
in Computational Mathematics*, vol.5, pp. 329–359,
1996.

[2] Corless, R.M, G.H. Gonnet, D.E.G. Hare, and D.J. Jeffrey
"Lambert's W Function in Maple" *The
Maple Technical Newsletter (MapleTech)*, vol.9, pp. 12–22,
1993.

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