e function and its properties

From: 3blue1brown

The function $e^t$ can be understood by examining its fundamental properties [00:00:00].

Defining Property

The most crucial, and arguably defining, property of the function $e^t$ is that it is its own derivative [00:00:08]. Coupled with the condition that an input of 0 yields an output of 1 ($e^0=1$), this makes $e^t$ the unique function possessing this characteristic [00:00:11].

Physical Intuition

This property can be illustrated using a physical model [00:00:17]:

• If $e^t$ represents your position on a number line over time, you start at position 1 [00:00:20].
• The equation implies that your velocity (the derivative of position) is always equal to your current position [00:00:24].
• This means the farther away from 0 you are, the faster you move [00:00:33].
• This relationship provides a strong intuitive understanding of how the function grows, indicating accelerating growth that feels “out of hand quickly” [00:00:42].

Variations of the Exponent

Constant Multiplier ($e^{kt}$)

If a constant $k$ is added to the exponent, such as $e^{2t}$, the chain rule dictates that the derivative becomes $k$ times itself [00:00:59].

• Positive Constant ($k>0$): For $e^{2t}$, the velocity is always twice the position. This leads to an even more rapid and “out of control” runaway growth [00:01:03]. At each point, instead of just attaching a vector corresponding to the position, you first double its magnitude [00:01:07].

• Negative Constant ($k<0$): If the constant is negative, like -0.5, the velocity vector is always negative 0.5 times the position vector [00:01:31]. This means the vector is flipped 180 degrees and its length is scaled by half [00:01:39]. This behavior results in movement in the opposite direction, slowing down in an exponential decay towards 0 [00:01:45].

Imaginary Constant ($e^{it}$)

When the constant is the imaginary unit $i$ (the square root of -1), as in $e^{it}$, the behavior shifts from the number line to the complex plane [00:01:57].

• The derivative of position will always be $i$ times itself [00:02:07].
• Multiplying a complex number by $i$ has the effect of rotating it 90 degrees [00:02:11].
• Therefore, the velocity at any point in time will be a 90-degree rotation of that position [00:02:25].
• Visualizing this for all possible positions creates a vector field [00:02:34].
• Starting at the initial condition $e^{i0}=1$ (at time $t=0$), the only trajectory where velocity consistently matches a 90-degree rotation of position is movement around a circle of radius 1 at a speed of 1 unit per second [00:02:42].
• This implies:
  • After $\pi$ seconds, $e^{i\pi }$ is -1 [00:03:01].
  • After $\tau$ (tau, or $2\pi$) seconds, $e^{i\tau }$ equals 1 (a full circle) [00:03:08].
  • More generally, $e^{it}$ represents a number that is $t$ radians around the unit circle in the complex plane [00:03:13].

Notational Considerations

The notation $e^t$ can be seen as a “notational disaster” because it places undue emphasis on the number $e$ and the concept of repeated multiplication, potentially obscuring the fundamental defining properties of the function itself [00:03:32].