Concept

Exponential

The function whose rate of growth equals its current value, written e to the power x. It turns addition into multiplication, describes compounding and decay, and on imaginary arguments it traces the unit circle.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A closer start buys time and nothing else. The logarithm of the separation between two Lorenz trajectories plotted against time, for three different initial separations. The three curves are straight and parallel over most of their length, with the same fitted slope.

A closer start buys only time

Two trajectories from almost the same place separate exponentially, and the rate does not depend on how close they began. Halving the initial error buys one fixed interval of extra agreement, and no amount of precision buys more than a fixed number of those.

dynamics · Strange attractor

A circular sector of area 1.10 and a hyperbolic sector of area 0.80. On the left, the unit circle with the sector from (1, 0) to (cos 2.2, sin 2.2), of area 1.100. On the right, the hyperbola x² − y² = 1 with the sector from (1, 0) to (cosh 1.6, sinh 1.6), of area 0.800. In both, the parameter is twice the shaded area.

The angle that is really an area

On a unit circle the angle t is the length of arc the point has walked, and it is also twice the area of the slice it has swept. The two readings agree on the circle and part company on the hyperbola — where arc length leads nowhere and area leads straight to cosh, sinh, and the exponential.

analysis · Circular functions

Named alongside it

The objects these essays reach for when they reach for this one.

Area Chaos Cosine Hyperbola Iteration Logarithm Lyapunov exponent Parametrisation Prediction Radian Sensitive dependence Sine

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