grammar

1 files changed, 1 insertions, 1 deletions

diff --git a/ClassJuly5.page b/ClassJuly5.page
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--- a/ClassJuly5.page
+++ b/ClassJuly5.page
@@ -22,7 +22,7 @@ You may find it helpful to think about other ways of deriving Theorems 1 and 2.
 
 An alternate proof of Theorem 2 goes as follows:  Since $f$ is holomorphic on a disk, it has a Laurent expansion.  The statement of Theorem 2 says that the negative terms in this Laurent expansion are zero.  First let's prove that $c_{-1}$ is zero.  Since $c_{-1}$ is the residue of $f$ at zero, it is given by
 $$c_{-1} = \int_{\gamma_r} f(z) dz$$
-where $\gamma$ is a small circle of radius $r$ that goes counterclockwise around the origin.  As we shrink the radius of this circle, its length goes to zero.  On the other hand since $f(z)$ tends to $f(0)$.  Taking the limit as $r \to 0$,
+where $\gamma$ is a small circle of radius $r$ that goes counterclockwise around the origin.  As we shrink the radius of this circle, its length goes to zero.  On the other hand since $f$ is holomorphic, $f(z)$ tends to $f(0)$ as $z$ tends to zero.  Taking the limit as $r \to 0$,
 $$ c_{-1} = \lim_{r \to 0} \int_{\gamma_r} f(z) dz = \lim_{r \to 0} 2 \pi r f(0) = 0  $$
 so we conclude that $c_{-1} = 0$.