Complex Analysis/Exercises/Sheet 2/Exercise 3

Task (Working with Polynomials, 5 Points)

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Consider a polynomial , given by

with and . Show that can also be expressed as a polynomial in and by specifying the coefficients in

.

Solution

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For ,is . Solving the system

for , we get and . We substitute these into . Using the Binomial theorem, we get

We then perform an index transformation, replacing the summation over with a summation over . Since , ranges from to . For a fixed , we consider values between and , where , so . Hence, ranges from to . We get

We then switch the order of summation. We have and for a fixed , , which means , resulting in

Finally, we perform another index transformation. We replace with . ranges from to . For a fixed , we have . Therefore, we consider only values of for which , i.e., . We get

Thus,

Translation and Version Control

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This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Übungen/2._Zettel/Aufgabe_3

  • Date: 01/14/2024