Deadly mutants are after us. What can give us hope?
This innocuous-looking function is a sublime light in the dark. It proves you can always recover. If your perseverance is infinite.
As x tends to zero, the exponent tends to minus infinity. The function’s value at zero tends to zero. It is a zero value of higher order: The function is infinitely flat.
Each and every derivative is also zero at zero, no matter the degree. When you differentiate the function, you get products of the exponential function and reciprocal powers of x. The exponential function does always win. It forces down the slope, the curvature, the change of the curvature, the change of changes of changes of infinite order.
But when all derivatives are zero – how can this function ever lift its head? How can it ever have a non-zero value anywhere? Haven’t we learned that functions are effectively power series? If we knew the value at one point and all the derivatives, we would know the function everywhere? As a physicist you learn – and quickly forget or ignore – that this is only true for ‘normal well-behaved’ functions.
But exp(-1/x^2) is not normal. At least not at x = 0.
Nobel prize winner Roger Penrose philosophizes about what is an honest function is, in his book The Road to Reality. An honest function is what mathematicians like Leonhard Euler would have appreciated. An honest function has no jumps or kinks; creating it must not include gluing together pieces of otherwise wildly different functions. A function has a chance to qualify as honest if represented by one simple formula – like ours.
But our function is not honest. It is not analytic, as for the lack of the power series about the zero point. In order to see a function’s true nature, Penrose tells us to consider it a complex-valued function of a complex argument (z = x + iy).
The plot above shows what happens if we traverse the zero point by following the real axis, changing the real value x and keeping the imaginary part zero. The function’s value is real, as it is if we follow the path along the imaginary axis (x=0). But now imaginary i is squared, and a crucial sign has changed:
Now the function’s value at zero seems to be infinity rather than zero.
Both the real-only path and the imaginary-only path lead to an absolute value of 1 when x or y tend to infinity. In the end everything is one.
But at the zero point (0 + 0i) the complex function looks pathological – both 0 and infinity at the same time. Or some other value. Travel along the line y=x through the zero point: Now the function’s absolute value is exactly 1 everywhere, maybe with the exception of the point 0+0i. When you follow different paths y=kx through the zero 0, with k’s from 0 to 1, the function’s absolute value is increasing until it reaches 1 for every point on the line y=x.
Moving further into the region where the imaginary part is larger than the real part, the absolute value rises rapidly from 1 to infinity.
But only a complex plot does this function’s magic justice. The complex function exp(-1/z^2) assigns a complex value to each complex argument z = x+iy. In order to display this 4-dimensional relationship in a flat 2-dimensional diagram, different hues represent different phase angles (arctan of the ratio of imaginary part and real part), and brightness values represent absolute values (sum of squares of real and imaginary part).
Close to zero points are either white to black – nearly infinity or nearly zero. At the diagonal line the absolute value is 1, and the function is continuous; yet it seems to jump from zero to infinity as changes in this exponential function are so steep.
The motivational function dances
around the central point of despair and weirdness.
Contour lines of constant magnitude or constant phase
like the number 8
or infinity ∞
if turned by ninety degrees.
The closer it gets to the strange point,
the more nervous become its ripples of changing phase.
The function can lift itself up,
from this infinitely flat valley.
When you walk around the flat valley,
when you turn by ninety degree,
you suddenly see a tall tower,
observation deck in infinite heights.
You can strike a balance,
and walk right through all this.
Everything is one all the time.