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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.

Figure 1. Exponential and logarithmic functions...Click to Enlarge Image

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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