The Euler e-function

The two sides of a power function $y=a{x}^n$ share a height, or take opposite signs, while the variable sits in the base. Put the variable in the exponent. For a real input the curve can still be drawn. Which writing of that curve also makes sense for a complex number, and makes the height at a sum equal the product of the heights? Sine and cosine of a complex input ask for a writing of the same kind. The exponent is the curve to settle first.

The curve passes through height 1

This picture keeps the input on the real line.

The fixed mark, at input $0$, has height $1$. Drag the input to the right and the height rises. Drag it to the left and the height falls toward the axis and stays positive. A complex definition has to reproduce this curve for every real input. This board stays on the real line, so a later shift by $2\pi i$ stays off the picture.

The series names every complex value

A complex number has the form $x=a+bi$ with $a$ and $b$ real, and the imaginary unit satisfies $i^2=-1$. Every real number is a complex number of that form. For every complex $x$,

$e^x:=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots ={\sum }_{k=0}^{\infty }\frac{x^k}{k!}.$

The sum starts at $k=0$, so the first term is $1$. That series is the Euler e-function. On the real line the partial sums are ordinary polynomials.

At input $0$ every partial sum has height $1$, the same height as the curve. Drag the degree upward and the mark at input $1$ climbs toward the curve.

Heights multiply when the inputs add

The two inputs on this board stay real. The mark at their sum lies on the curve, and its height is the product of the two heights. The identity that matches the picture holds for every complex $x$ and $z$:

$e^{x+z}=e^x{e}^z.$

Set the input equal to $1$. The series names the number

$e:=1+1+\frac{1}{2!}+\frac{1}{3!}+\cdots ,$

and $e=2.71828183$.

For every real $x$ the same function has a second writing. The extra assumption is that the input is real.

$e^x={\lim }_{n\to \infty }{\left(1+\frac{x}{n}\right)}^n.$

Drag the count upward. The moving mark climbs toward the height of the curve at that real input. At input $1$ the height it approaches is $e$.

It outgrows every power

On the real line the Euler e-function is continuous and strictly increasing: a larger input gives a larger height. Far to the right the height grows without bound. Far to the left it falls toward $0$, the tail already visible on the first board.

${\lim }_{x\to +\infty }{e}^x=+\infty ,{\lim }_{x\to -\infty }{e}^x=0.$

For each integer $n=1,2,\dots$,

${\lim }_{x\to +\infty }\frac{e^x}{x^n}=+\infty .$

When $n\ge 2$, $x^n$ is a power function with the coefficient $1$. The height $e^x$ eventually overtakes that power, and the same limit already holds for $n=1$. If a count $N$ makes the work grow like $e^N$, that work becomes unusable as $N$ increases.

The slope equals the height

For every real or complex $x$ the Euler e-function is infinitely differentiable, and its derivative equals itself:

$\frac{\mathrm{d}{e}^x}{\mathrm{d}x}=e^x.$

The board keeps the input real. One step to the right along the tangent rises by the height at the point where the step began.

At input $0$ the height is $1$, and the tangent reaches height $2$ one step to the right. Drag the input and the vertical step stays equal to the height.

A shift by $2\pi i$ repeats

For every complex $x$,

$e^{x+2\pi i}=e^x.$

That period leaves the real line. The real graph is one rising curve. The value stays off zero for every complex $x$,

$e^x\ne 0,$

and on the real line the first board already keeps the height positive.

The series defines the Euler e-function for every complex number, and every real number is one of those inputs. On the real line the graph passes through height $1$, rises as the input grows, falls toward the axis on the left, and has slope equal to its height. Adding $2\pi i$ returns the same value, and that real graph is a single curve with no zero.