# Equivalence of Definitions of Complex Exponential Function/Power Series Expansion equivalent to Definition by Real Functions

Jump to navigation
Jump to search

## Theorem

The following definitions of the concept of **Complex Exponential Function** are equivalent:

### As a Power Series Expansion

The **exponential function** can be defined as a (complex) power series:

\(\ds \exp z\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 1 + \frac z {1!} + \frac {z^2} {2!} + \frac {z^3} {3!} + \cdots + \frac {z^n} {n!} + \cdots\) |

### By Real Functions

The **exponential function** can be defined by the real exponential, sine and cosine functions:

- $\exp z := e^x \paren {\cos y + i \sin y}$

where $z = x + i y$ with $x, y \in \R$.

Here, $e^x$ denotes the real exponential function, which must be defined first.

## Proof

We have the result:

which gives that the definition of $\exp z$ as the power series:

- $\exp z := \ds \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}$

is equivalent to the definition of $\exp z$ as the solution of the differential equation:

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

satisfying the initial condition $y \paren 0 = 1$.

Let:

- $e: \R \to \R$ denote the real exponential function
- $\sin: \R \to \R$ denote the real sine function
- $\cos: \R \to \R$ denote the real cosine function.

Then:

\(\ds \exp z\) | \(=\) | \(\ds \map \exp {x + i y}\) | where $x, y \in \R$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \exp x \map \exp {i y}\) | Exponential of Sum: Complex Numbers | |||||||||||

\(\ds \) | \(=\) | \(\ds e^x \map \exp {i y}\) | Definition (as Power Series Expansion) agrees with Definition of Real Exponential for all $x \in \R$ | |||||||||||

\(\ds \) | \(=\) | \(\ds e^x \paren {\cos y + i \sin y}\) | Euler's Formula, which can be proven using Definition as Power Series Expansion |

$\blacksquare$

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$