Sum of Complex Indices of Real Number

Theorem

Let $r \in \R_{> 0}$ be a (strictly) positive real number.

Let $\psi, \tau \in \C$ be complex numbers.

Let $r^\lambda$ be defined as the the principal branch of a positive real number raised to a complex number.

Then:

$r^{\psi \mathop + \tau} = r^\psi \times r^\tau$

Proof

Then:

\(\ds r^{\psi \mathop + \tau}\)

\(=\)

\(\ds \map \exp {\paren {\psi + \tau} \ln r}\)

Definition of Principal Branch of Positive Real Number raised to Complex Number

\(\ds \)

\(=\)

\(\ds \map \exp {\psi \ln r + \tau \ln r}\)

\(\ds \)

\(=\)

\(\ds \map \exp {\psi \ln r} \, \map \exp {\tau \ln r}\)

Exponential of Sum

\(\ds \)

\(=\)

\(\ds r^\psi \times r^\tau\)

Definition of Principal Branch of Positive Real Number raised to Complex Number

$\blacksquare$