conjugateComplex

ConjugateComplex refers to the complex conjugate of a complex number, or to a function that computes this conjugate in mathematical or programming contexts. If a complex number is written as z = a + bi, where a and b are real and i is the imaginary unit, then its complex conjugate is z̄ = a − bi. The operation leaves the real part unchanged and negates the imaginary part. For real numbers, z̄ = z; for purely imaginary numbers z = bi, z̄ = −bi.

Key properties include:

- z z̄ = |z|^2, a nonnegative real number.

- (z̄)̄ = z.

- (zw)̄ = z̄ w̄ and (z + w)̄ = z̄ + w̄.

- If z ≠ 0, 1/z = z̄ / |z|^2.

In programming, a function named conjugateComplex typically returns the complex conjugate of its input. Representations vary: inputs may be pairs (a, b) or a built-in complex number type, with the result as (a, −b). When operating on arrays or matrices of complex numbers, the conjugate is usually applied element-wise; the conjugate transpose (Hermitian transpose) combines conjugation with transposition.

Examples:

- z = 3 + 4i yields z̄ = 3 − 4i.

- z = 5 yields z̄ = 5.

- z = −2i yields z̄ = 2i.

Applications include simplifying division of complex numbers, computing Fourier and Laplace transforms, and contexts in physics and engineering where conjugation underpins inner products and Hermitian operators. See also complex conjugate, modulus, complex plane, and conjugate transpose.