Is this proof correct? Let $D$ be a domain. By the theorem of existence of harmonic conjugates, we get that every harmonic function on $D$ has a harmonic conjugate.
In other words, for every harmonic function $u$ on $D$, there exists a harmonic function $v$ on $D$ such that $f=u+i\,v$ is analytic on $D$. Since $f$ is analytic on $D$, so $f$ is infinitely (complex) differentiable. It follows that $u$ and $v$ are infinitely differentiable. In particular, this show that $u$ has partial derivatives of all orders.
In other words, for every harmonic function $u$ on $D$, there exists a harmonic function $v$ on $D$ such that $f=u+i\,v$ is analytic on $D$. Since $f$ is analytic on $D$, so $f$ is infinitely (complex) differentiable. It follows that $u$ and $v$ are infinitely differentiable. In particular, this show that $u$ has partial derivatives of all orders.
I'm reading a proof of linear recurrence relation with constant coefficients of order $k$, which gives the formula of $a_n$ when some roots of its characteristic equation has multiplicity $\ge2$ . In one step it says:
\begin{align*}
&\textrm{Since }\alpha_i\textrm{ satisfy}\\
&C_n\alpha^n+C_{n-1...
I ran across the following math problem where there is arithmetic involved:
$$9-4+1$$
Where supposedly the answer is 6? I entered into my computer and calculator and got the same result so I realized there was something strange in this math problem because it does not follow the PEMDAS pattern.
...
Show that if the power series $\sum_{n=0}^\infty a_n x^n$ has radius of convergence $R$ and if $\lim_{n \to \infty} |a_{n+1}/a_n|$ exists, then the value of this limit is $R$
I think there might be a typo in the problem: I think the conclusion should be that $1/R$ is the limit of the sequenc...
