# 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.[1]
2. The integral over this curve can then be computed using the residue theorem.[2]
3. The new winding number allows to establish a generalized residue theorem which covers also the situation where singularities lie on the cycle.[3]
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.[3]
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 .[3]
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).[3]
7. The Espil's theorem it's a short proof of the Cauchy's generalized residue theorem.[4]
8. However, I decided to use the nuclear bomb of the integration arsenal, the Cauchy residue theorem of complex analysis.[5]
9. In an upcoming topic we will formulate the Cauchy residue theorem.[6]
10. 1 Residue theorem problems 2 2 Zero Sum theorem for residues problems 76 3 Power series problems 157 Acknowledgement.[6]
11. The following result, Cauchy’s residue theorem, follows from our previous work on integrals.[6]
12. Using residue theorem to compute an integral.[6]

## 메타데이터

### Spacy 패턴 목록

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