diff options
Diffstat (limited to 'mindmap')
| -rw-r--r-- | mindmap/Cauchy's Theorem.org | 4 | ||||
| -rw-r--r-- | mindmap/limit.org | 2 |
2 files changed, 2 insertions, 4 deletions
diff --git a/mindmap/Cauchy's Theorem.org b/mindmap/Cauchy's Theorem.org index 01d507a..38f6c06 100644 --- a/mindmap/Cauchy's Theorem.org +++ b/mindmap/Cauchy's Theorem.org @@ -19,9 +19,7 @@ functions. For example: and many more, are statements about the same set of objects, posed in different ways. * Theorem #+begin_theorem -\begin{align} -\oint_{D}f(z)dz = 0 -\end{align} +If $D$ is a bounded domain with piecewise smooth boundary and $f$ is an analytic function which extends smoothly to $D \cup \partial D$, then $\oint_{D}f(z)dz = 0$. #+end_theorem #+begin_proof diff --git a/mindmap/limit.org b/mindmap/limit.org index e8ef2ea..d90eabf 100644 --- a/mindmap/limit.org +++ b/mindmap/limit.org @@ -95,7 +95,7 @@ in every neighbourhood of some $x \in X$, and thus converges. #+end_proof and finally we get the Bolzano-Weierstrass theorem for $\mathbb{R}^{n}$ for free, as $\mathbb{R}^{n} \cup \lbrace \infty\rbrace$ is a compact metric space: #+begin_corollary -Every sequence in $R^{n}$ either has a convergent subsequence, or has a subsequence that escapes to $\infty$. +Every sequence in $\mathbb{R}^{n}$ either has a convergent subsequence, or has a subsequence that escapes to $\infty$. #+end_corollary Also, for the two-point compactification, it yields this result as well if you're working in that space: #+begin_corollary |