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The Jacobian determinant of a holomorphic map is and is positive exactly where
Statement
Let be holomorphic on an open set . At every ,
The determinant is positive exactly where , and it is zero exactly where ; it is never negative.
Facts & Assumptions
Given: A holomorphic map and a point in its open domain.
If , then the real derivative matrix is (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with , or with the Cauchy–Riemann equations).
The Jacobian matrix is the matrix of the first partial derivatives (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).
Complex modulus satisfies , and if and only if (Conjugation is an involutive real-field automorphism, , and modulus is definite, multiplicative, and subadditive).
Proof
By [L1] and [F1], .
The sum is nonnegative and, by [L2], is zero exactly when ; otherwise it is positive.
Depends on
Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with , or with the Cauchy–Riemann equations
The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
Conjugation is an involutive real-field automorphism, , and modulus is definite, multiplicative, and subadditive