University:
Massachusetts Institute of Technology
Course:
18.04 | Complex Variables with Applications
Academic year:

2020
Views:

377

Pages:

1
Author:

Miya Brewer

18.04 Recitation12 1. Use Rouche’s theorem to show that all 5 zeros of �5 + 3� + 1 are inside the curve �2 ∶= {� ∶ |�| = 2}. Ans: Example 11.7 in the notes. 2. Use Rouche’s theorem to show that � + 3 + 2�� has one zero in the left half plane. Ans: Example 11.8 in the notes. 3. Use Rouche’s theorem to give another proof of the fundamental theorem of algebra i.e. to show that �� + ��−1 ��−1 + ⋯ + �0 has exactly � roots in the complex plane. Ans: Theorem following Example 11.8 in the notes. �(�) 4. Let �(�) be a meromorphic function, and let �(�) ∶= 1+�(�) . For a closed curve � such that �◦� does not go through −1 and � does not have any poles on �, show that ��,� = ��,� + Ind(�◦�, −1). Ans: Note that ��,� = �1+�,� , so it suﬃces to show that �1+�,� − ��,� = Ind(�◦�, −1). Since 1 + � does not have zeros or poles on �, the argument principle shows that �1+�,� − �1+�,� = Ind((1 + �)◦�, 0) = Ind(1 + �◦�, 0) = Ind(�◦�, −1). Finally, note that �1+�,� = ��,� . 5. Consider the method of images if there is a source at (0, 0), and walls at � = 1 and � = −1. How many image sources do you need? What is the resulting complex potential? Ans: We add an image source at (0, 2) to make the � = 1 wall a streamline. Then, we add image sources at (0, −2) and (0, −4) to make the � = −1 wall a streamline. However, when we do so, the � = 1 wall is no longer a streamline, so we need to add sources at (0, 4) and (0, 6). When we do this, the � = −1 wall is no longer a streamline, and so on. Continuing this process, we see that we need a source at ±(0, 2�) for all � ≥ 1, and the resulting complex potential is ∑ ∑ Φ(�) = log(�) + log(� + 2��) + log(� − 2��). �≥1 �≤−1