COMPUTING SCIENCE
Why W?
Brian Hayes
Single U and W
In trying to make sense of the expression WeW ,
the first question is not "why W?" but "why
e?" The e in the formula is Euler's number,
the second-most-famous constant in all of mathematics, often
introduced as "the base of the natural logarithms"; but
then the natural logarithms are usually defined as "logarithms
taken to the base e," which is not much help. Another
way of defining e, originally derived from the study of
compound interest, avoids this circularity: e is the
limiting value of the expression (1+1/n)
n as n tends to infinity. Thus we can
approximate the value of e by setting n equal to
some arbitrary, large number. With n=1,000,000, for
example, we get six correct digits of e: 2.71828.
Now, with e in hand, consider an equation somewhat
simpler than the one for W; we might call it single-U:
eU = x.
This equation defines the exponential function, also written
exp(U). The function maps each given value of U to
a corresponding value of x, namely e raised to the
power U. If U is a positive integer, we can
calculate the function's value by simple arithmetic: Just multiply
e by itself U times. For nonintegral U,
the procedure is not quite so obvious but is still well-defined; the
left part of Figure 1 shows the elegant curve generated.
The equation eU = x also defines an
inverse function; we just need to read the equation backward.
Whereas the forward function maps a value of U to a value
of x, the inverse function takes a value of x as
input and returns the corresponding value of U. In other
words, the inverse function finds the power to which e must
be raised to yield a given value of x. This is another
well-known, textbook function: the natural logarithm, written
log(x) or ln(x). The log function has the
same graph as the exponential function but reflected across the
diagonal, as shown in the right part of Figure 1.
Sometimes it's helpful to think of functions like these as if they
were machines. The exp function works like a meat grinder:
Dump a U into the input hopper, turn the crank, and out
comes an x equal to eU . The same
machine can calculate logarithms if we run numbers through it
backwards and turn the crank the other way, but there is an
important caveat: When the machine runs in reverse, some inputs can
jam the gears. For example, what output should the machine produce
if you ask it for the logarithm of 0 or -1? (If you have a
scientific calculator handy, see how a real machine answers these questions.)
The problem is that the logarithm function works only over a limited
domain. Exponentiation is defined over the entire real number line;
any real value of U, whether positive or negative, produces
a value of exp(U), and that value is always a positive real
number. The inverse function is not so well-mannered: As the right
side of Figure 1 suggests, log(x) is defined only if
x is positive. This limitation can be sidestepped by
venturing off the real number line into the wilds of the complex
plane. If the value of log(x) is allowed to be a complex
number, with both a real and an imaginary part, then
log(-1) has a definite value: According to a famous formula
of Euler, it is equal to π i, where i is the
imaginary unit, the square root of -1. The W function is
also defined throughout the complex plane, but in this article I
shall confine myself to the straight and narrow path of real
numbers. However, see Figure 4.
One more simple fact about logarithms will be needed below. The
logarithm of the product of two numbers is equal to the sum of the
logarithms of the factors: log(xy)=log(x)+log(y). Likewise
for quotients: log(x/y)=log(x)-log(y). These relations were
the main reason for inventing logarithms in the first place: They
convert multiplication and division into the easier tasks of
addition and subtraction.
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