# Exponential of Sum/Complex Numbers

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## Theorem

Let $z_1, z_2 \in \C$ be complex numbers.

Let $\exp z$ be the exponential of $z$.

Then:

- $\map \exp {z_1 + z_2} = \paren {\exp z_1} \paren {\exp z_2}$

### Corollary

Let $z_1, z_2 \in \C$ be complex numbers.

Let $\exp z$ be the exponential of $z$.

Then:

- $\map \exp {z_1 - z_2} = \dfrac {\exp z_1} {\exp z_2}$

### General Result

Let $m \in \N_{>0}$ be a natural number.

Let $z_1, z_2, \ldots, z_m \in \C$ be complex numbers.

Let $\exp z$ be the exponential of $z$.

Then:

- $\ds \map \exp {\sum_{j \mathop = 1}^m z_j} = \prod_{j \mathop = 1}^m \paren {\exp z_j}$

## Proof

This proof is based on the definition of the complex exponential as the unique solution of the differential equation:

- $\dfrac \d {\d z} \exp = \exp$

which satisfies the initial condition $\map \exp 0 = 1$.

Define the complex function $f: \C \to \C$ by:

- $\map f z = \map \exp z \, \map \exp {z_1 + z_2 - z}$

Then find its derivative:

\(\ds D_z \, \map f z\) | \(=\) | \(\ds \paren {D_z \, \map \exp z} \map \exp {z_1 + z_2 - z} + \map \exp z \paren {D_z \map \exp {z_1 + z_2 - z} }\) | Derivative of Complex Composite Function | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \exp z \, \map \exp {z_1 + z_2 - z} + \map \exp z \, \map \exp {z_1 + z_2 - z} \map {D_z} {z_1 + z_2 - z}\) | as $\exp$ is its own derivative | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \exp z \, \map \exp {z_1 + z_2 - z} - \map \exp z \, \map \exp {z_1 + z_2 - z}\) | Derivative of Complex Power Series | |||||||||||

\(\ds \) | \(=\) | \(\ds 0\) |

From Zero Derivative implies Constant Complex Function, it follows that $f$ is constant.

Then:

\(\ds \map \exp {z_1} \, \map \exp {z_2}\) | \(=\) | \(\ds \map f {z_1}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map f 0\) | as $f$ is constant | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \exp 0 \, \map \exp {z_1 + z_2}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map \exp {z_1 + z_2}\) | as $\map \exp 0 = 1$ |

$\blacksquare$

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(i)}$: $(4.12)$ - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $3$ - 2001: Christian Berg:
*Kompleks funktionsteori*: $\S 1.5$