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### Solution to the Helmholtz equation with a specific form

I want to find a function $f$ that solves the Helmholtz equation $$\Delta_w f(x,w)+\lambda f(x,w)=0,$$ and satisfy the natural Neumann boundary condition. However, I am looking for a specific form ...
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### Let $f_1:A \rightarrow B$, $f_2:C \rightarrow B$ functions. Prove that, if $f_1(x)=f_2(x)$ for all $x\in A\cap C$, then $f_1 \cup f_2$ is function.

I want to test the above, so my idea is as follows: take $f=f_1 \cup f_2$ and prove that $f:A\cap C \rightarrow B$ is function. To do this, we start with $x \in A\cap C$, then $x \in A$ and $x \in C$, ...
• 633
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### prime factors of $n=x^2+y^2$

Let $n$ be a positive integer. Show that there exist integers $x$ and $y$ such that $n=x^2+y^2$ if and only if each prime factor of $n$ of the form $4k+3$ appears an even number of times
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