# 유수 정리 (residue theorem)

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## 응용

$\sum_{k=1}^{\infty}\frac{1}{k^{4}-a^4}=\frac{1}{2a^4}-\frac{\pi \cot (\pi a)}{4 a^3}-\frac{\pi \coth (\pi a)}{4 a^3}$ $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+n+1}=\frac{2\pi \tanh \left(\frac{\sqrt{3} \pi }{2}\right)}{\sqrt{3}}$

## 수학용어번역

• residue - 대한수학회 수학용어집

## 노트

### 말뭉치

1. Applying the Cauchy residue theorem.
2. The integral over this curve can then be computed using the residue theorem.
3. The new winding number allows to establish a generalized residue theorem which covers also the situation where singularities lie on the cycle.
4. This residue theorem can be used to calculate the value of improper integrals for which the standard technique with the classical residue theorem does not apply.
5. In the present article, we introduce a generalized, non-integer winding number for piecewise cycles and a general version of the residue theorem which covers all cases of singularities on .
6. Definition 2 of a generalized winding number turns out to be useful as it allows to generalize the residue theorem (see Theorem 8 below).
7. The Espil's theorem it's a short proof of the Cauchy's generalized residue theorem.
8. However, I decided to use the nuclear bomb of the integration arsenal, the Cauchy residue theorem of complex analysis.
9. In an upcoming topic we will formulate the Cauchy residue theorem.
10. 1 Residue theorem problems 2 2 Zero Sum theorem for residues problems 76 3 Power series problems 157 Acknowledgement.
11. The following result, Cauchy’s residue theorem, follows from our previous work on integrals.
12. Using residue theorem to compute an integral.

## 메타데이터

### Spacy 패턴 목록

• [{'LOWER': 'residue'}, {'LEMMA': 'theorem'}]
• [{'LOWER': 'cauchy'}, {'LOWER': 'residue'}, {'LEMMA': 'theorem'}]
• [{'LOWER': 'cauchy'}, {'LOWER': "'s"}, {'LOWER': 'residue'}, {'LEMMA': 'theorem'}]