Math and stuff

Proposition If $A$ is absolutely flat, every primary ideal is maximal. Solution Let $q$ be a primary ideal of $A$. It suffices to show that $A / q$ is a field. By the definition of a primary ideal, $A / q \ne 0$. Let $x \in A \setminus q$. By...

Proposition Is it possible to find a real function $v(x, y)$ so that $x^3 + y^3 + iv(x, y)$ is holomorphic? Solution By Proposition 6.3 [A first course in complex analysis], the real and imaginary parts of a holomorphic function must be harmonic. $u(x, y) = x^3 + y^3$ is...

Proposition Consider $u(x, y) = \ln(x^2 + y^2)$. Show that $u$ is harmonic on $\mathbb{C} \setminus \{ 0 \}$. Prove that $u$ is not the real part of a function that is holomorphic in $\mathbb{C} \setminus \{ 0 \}$. Solution 1 $u_x = 2x / (x^2 + y^2)$ and $u_y...

Proposition Let $u(x, y) = e^x\sin y$. Show that $u$ is harmonic on $\mathbb{C}$. Find an entire function $f$ such that $\Re(f) = u$. Solution 1 \(u_{xx} = e^x\sin y\) and \(u_{yy} = -e^x\sin y\). Therefore, \(u_{xx} + u_{yy} = 0\). 2 We will use Theorem 6.8[A first course in...

Proposition Show that every non-zero $\omega \in \Lambda^n(V)$ is the volume element determined by some inner product $T$ and orientation $\mu$ for $V$. Solution Let $v_1, \cdots, v_n$ be a basis of $V$. Suppose $\omega(v_1, \cdots, v_n) = 0$. Then $\omega(v_{\sigma_1}, \cdots, v_{\sigma_n}) = 0$ for all $\sigma \in S_n$....