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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2002 • Question 8

Question

Math 6210 • Real Analysis 2002 • Question 8

Let \(f\) be a complex valued function defined on the domain \(D\subset \mathbb C\). Suppose, that as a mapping of \(\mathbb R^2\) to \(\mathbb R^2\), \(f=u+iv\) is twice differentiable, and that the jacobian \[ \begin {pmatrix} \dfrac {\partial u}{\partial x} & \dfrac {\partial u}{\partial y}\\[0.6ex] \dfrac {\partial v}{\partial x} & \dfrac {\partial v}{\partial y} \end {pmatrix} \] is everywhere, as a linear transformation, a composition of a dilation and a rotation. Show that \(f\) is a holomorphic function.

Raw LaTeX

Let \(f\) be a complex valued function defined on the domain
\(D\subset \mathbb C\). Suppose, that as a mapping of \(\mathbb R^2\) to
\(\mathbb R^2\), \(f=u+iv\) is twice differentiable, and that the jacobian
\[
\begin{pmatrix}
\dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y}\\[0.6ex]
\dfrac{\partial v}{\partial x} & \dfrac{\partial v}{\partial y}
\end{pmatrix}
\]
is everywhere, as a linear transformation, a composition of a dilation and a
rotation. Show that \(f\) is a holomorphic function.

Topic Tags

Almost Everywhere and Convergence in Measure Automorphism Groups and Biholomorphisms Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms