I've recently been working with the Lambert W function. I think I might have found a way to calculate the W function, but I also think the way probably contains an ommission of some sort. Anyways, here's the proof.

The definition of the W function is

y=W(z) if ye^y =z. Let y=ix.

ixe^ix =ix(cos(x)+isin(x))=-xsin(x)+ixcos(x)

Let z=a+ib,

a=-xsin(x)
b=xcos(x)

Then,

a² +b² =x² sin² (x)+x² cos² (x)

=x² (sin² (x)+cos² (x))

=x²

Thus

x=±(a² +b² )^1/2 =±|z|

So, y=W(z)=±i|z|. Of course this proof assumes that W(z) has a complex solution. If W(z) has a complex solution, it is given by ±i|z|, though. There are two problems I've noticed so far.

a) This doesn'T seem like it would have not been noticed.

and b) It's rather late, even here in America.

Other than these though, I don't really see a flaw. Is there one?

Thanks,

Brad

However, if there is a pure imaginary solution to $y{e}^{y}z$ (i.e. $\mathrm{Re}(y)=0$) then your method will find it because then $x$ will be a real number. I don't think this will happen very often though, because it requires that $i\left|z\right|e^{i\left|z\right|=z}$ which happens if and only if $\arg (z)=\pi /2+\left|z\right|$. You could actually plot this using 2d polar coordinates as $r=\theta -\pi /2$ for $\theta >\pi /2$. It's just a spiral. In other words, the set of complex points for which your formula works is a spiral. Here's what it looks like:

I follow you on most of that, but why shouldn't a and b be real if we explicitly define a=Re(z) and b=Im(z)? Are they quaternions? Also, if the series for sine and cosine are defined for the whole complex plane, why shouldn't sin² x+cos² x=1 even for complex x?

Is there any other information about the W function that is known?

One more: are there 'z's such that W(z) does not exist? (Or at least is not a complex number)

Thanks,

Brad

Brad, we could explicity define a=Re(z) and b=Im(z), but then it would be wrong to say that a=-x.sin(x) and b=x.cos(x) because x need not be a real number. It is true that sin² z+cos² z=1 for all z in the complex plane, but this still doesn't prove what you wanted. Does that explain it? I don't know much more about the W (sometimes called product logarithm) function I'm afraid.

It is easy to seperate

(x+iy)e^x+iy =a+ib

into real and imaginary parts, and it comes out to be

a=xcos(y)-ysin(y)
b=xsin(y)+ycos(y)

An interesting consequence of this is that for a given y,

[(da/dx)² +(db/dx)² ]=1

So x is equal to the length of a curve generated by putting y at it's proper value.

I would doubt that the two equations giving values for a and b above can be solved, but could a series expansion be made out of them?

Hope that helps.