---
tag: 019H1Z6419
kind: statement
role: proposition
name: Derivative of the complex exponential
msc: [33B10]
uses: [054VXHMCA3, 0D1C9Y1EXA]
sources:
  - url: https://en.wikipedia.org/wiki/Exponential_function
    locator: Complex exponential
  - url: https://en.wikipedia.org/wiki/Exponential_function
    locator: Differential equation
---

## Statement

For each $z \in \mathbb{C}$, the [[complex exponential|054VXHMCA3]] is [[complex differentiable|0D1C9Y1EXA]] at $z$, and
$$\exp'(z) = \exp z.$$

## Remarks

Proof: differentiate the series $\sum_{n \ge 0} z^n / n!$ term by term. The derivative of $z^n / n!$ is $z^{n-1} / (n-1)!$ for $n \ge 1$, so the series of derivatives is the same series.
With the chain rule, the derivative of $e^{a z}$ is $a e^{a z}$ for each constant $a \in \mathbb{C}$.
It follows that $\exp$ is [[entire|0G2YJN0H5S]].

## Explanations

### L1 human

As in the real case, the exponential function is its own derivative.
You can see it in the power series: differentiating each term gives the term before it, so the whole series does not change.