Derivative of Exponential Function/Complex
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Theorem
The complex exponential function is its own derivative.
That is:
- $\map {D_z} {\exp z} = \exp z$
Proof from Sequence Definition
Take the definition of $\exp$ to be the limit of the sequence $\left \langle{E_n}\right \rangle$ defined by:
- $\displaystyle E_n \left( {z} \right) = \left({1 + \dfrac z n}\right)^n$
Then $\left \langle{E_n}\right \rangle$ is uniformly convergent on compact subsets of $\C$.
Further, $\C$ is an open, connected subset of $\C$.
So the hypotheses of Derivative of Sequence of Holomorphic Functions are satisfied.
Hence:
| \(\displaystyle D_z \left({\exp z}\right)\) | \(=\) | \(\displaystyle D_z \left({1 + \dfrac z n}\right)^n\) | Definition of Complex Exponential Function | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \lim_{n \to \infty} D_z \left({1 + \dfrac z n}\right)^n\) | Derivative of Sequence of Holomorphic Functions | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \lim_{n \to \infty} \left({ n \left({1 + \dfrac z n}\right)^{n - 1} \times \frac 1 n}\right)\) | Chain Rule | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \lim_{n \to \infty} \left({1 + \dfrac z n}\right)^{n - 1}\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \lim_{n \to \infty} \left({ \left({1 + \dfrac z n}\right)^n \times \frac n {n + z} }\right)\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left({ \lim_{n \to \infty} \left({1 + \dfrac z n}\right)^n }\right) \times \left({ \lim_{n \to \infty} \frac n {n + z} }\right)\) | Complex Derivative of Product is Product of Complex Derivative | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \exp z \times 1\) | Definition of Complex Exponential Function |
Hence the result.
$\blacksquare$
