Exponential Form of Complex Conjugate

Theorem

Let $z := r e^{i \theta} \in \C$ be a complex number expressed in exponential form.

Then:

$\overline z = r e^{-i \theta}$

where $\overline z$ denotes the complex conjugate of $z$.

Proof

\(\ds z\)

\(=\)

\(\ds r e^{i \theta}\)

\(\ds \)

\(=\)

\(\ds r \paren {\cos \theta + i \sin \theta}\)

Euler's Formula

\(\ds \leadsto \ \ \)

\(\ds \overline z\)

\(=\)

\(\ds r \paren {\cos \theta - i \sin \theta}\)

Polar Form of Complex Conjugate

\(\ds \)

\(=\)

\(\ds r e^{-i \theta}\)

Corollary to Euler's Formula

$\blacksquare$

Sources

• 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.3$ Complex conjugation