diff options
| -rw-r--r-- | ClassJuly5.page | 4 |
1 files changed, 3 insertions, 1 deletions
diff --git a/ClassJuly5.page b/ClassJuly5.page index abebe9e..2e9314d 100644 --- a/ClassJuly5.page +++ b/ClassJuly5.page @@ -14,7 +14,9 @@ The reason we discussed convergence of Fourier series was to give some taste for The reason we discussed the Heat and Wave equations was to illustrate other examples of the methods we used to prove Theorems 1 and 2. So, if you only care about holomorphic functions you don't need to worry about those examples. -You may find it helpful to think about other ways of deriving Theorems 1 and 2. For an alternate proof of Theorem 1 (which may be more comprehensible, since it doesn't involve changing coordinates), see Problems 7 and 8 of Problem Set 3. +## Alternate Proofs + +You may find it helpful to think about other ways of deriving Theorems 1 and 2. For an alternate proof of Theorem 1 (which may be more comprehensible, since it doesn't involve any confusing changes of coordinates), see Problems 7 and 8 of Problem Set 3. 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 $$c_{-1} = \int_{\gamma_r} f(z) dz$$ |