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

From ProofWiki
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:

Power Series Expansion equivalent to Solution of Differential Equation

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