COMPUTING SCIENCE
Why W?
Brian Hayes
W Coming and Going
There is an obvious family resemblance between single-U and
W, between the equations eU =
x and WeW = x. In the case of
the forward W function, if we know how to calculate
eW , then it's a trivial matter to calculate
WeW : just multiply by W. The resulting
curve is shown in the left part of Figure 2. In overall shape it
looks much like the exponential curve, although for large W
it rises more steeply. Where eW and
WeW really part company is to the left of
W=0. Whereas eW is always positive,
WeW dips into negative territory, reaching a
minimum at the point W=-1, x=-1/e. As
W tends toward negative infinity, both
eW and WeW approach 0, but
one from above and the other from below.
Taking the inverse of this function—solving
WeW = x for W instead of for
x—finally brings us to the Lambert W
function. Just how to solve for W is a matter I'll
return to below, but for now it's enough to flip the graph of the
function about its diagonal, as in the right side of Figure 2; the
inverse graph is drawn in more detail in Figure 3. Just as the
forward function resembles the exponential curve, the inverse
function appears similar to the logarithm. The curves for
log(x) and W(x) cross at x=e, where both
are equal to 1. Where things get most interesting, again, is to the
left of x=0. Whereas log(x) is undefined for any
x ≤ 0, W(x) continues to have a value down to
x=-1/e, or about -0.37. Indeed, when x
lies in the range between -1/e and 0, W(x) has not
just a value but two values. For example,
W(-0.2) could be equal to either -0.26 or -2.54. Plugging
either of these W values into the formula
WeW yields the x value -0.2.
For a mathematical function, multiple values are an embarrassment of
riches; a well-bred function is supposed to map each value in its
domain to a single value in its range. But in practice multiple
values are not uncommon, particularly with inverse functions. The
square root is a familiar example: Whereas squaring 2 yields the
unique result 4, the square root of 4 could be either +2 or -2. Some
of the trigonometric functions are even worse. Every angle has just
one sine, but the inverse function, the arc sine, wraps around to
produce infinitely many values.
The problem with multivalued functions is knowing which value, or
branch, to choose. Most calculators and programming languages give
precedence to positive roots and to arc sine values between -90 and
+90 degrees, but there is no fundamental justification for these
choices. In the case of Lambert W, the part of the curve
with W>-1 has been labeled the "principal
branch," but again this is mainly a matter of convention. (In
the complex plane, W has infinitely many branches.)
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