281 Complex Analysis
definition [D] (Complex Conjugate) A unique dual complex number associated with a given complex number, geometrically representing its reflection across the real axis in the complex plane, where the following condition applies:
- For a complex number \(z = x + iy\), the conjugate \(z^*\) is \(x - iy\).
where
- \(z\) is a complex number composed of real and imaginary parts.
- \(z^*\) is the complex conjugate.
- \(x\) is the real part, denoted \(\text{Re}(z)\).
- \(y\) is the imaginary part, denoted \(\text{Im}(z)\).
- \(i\) is the imaginary unit satisfying \(i^2 = -1\).
Note:
- \(z^*\) is also written \(\bar{z}\).