"유수 정리 (residue theorem)"의 두 판 사이의 차이
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| − | == | + | ==개요== |
| + | * [[복소함수론]]의 주요 정리 중 하나 | ||
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| − | + | ==응용== | |
| − | + | * [[데데킨트 합]] | |
| − | + | * [[왓슨 변환(Watson transform)]] | |
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| − | * [[데데킨트 합]] | ||
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| − | * [[왓슨 변환(Watson transform)]] | ||
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| − | + | :<math>\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}</math> | |
| + | :<math>\sum_{n=-\infty}^{\infty}\frac{1}{n^2+n+1}=\frac{2\pi \tanh \left(\frac{\sqrt{3} \pi }{2}\right)}{\sqrt{3}}</math> | ||
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| + | ==역사== | ||
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
| − | * [[ | + | * [[수학사 연표]] |
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==메모== | ==메모== | ||
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* http://www.math.binghamton.edu/sabalka/teaching/09Spring375/Chapter10.pdf | * http://www.math.binghamton.edu/sabalka/teaching/09Spring375/Chapter10.pdf | ||
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==관련된 항목들== | ==관련된 항목들== | ||
| 67번째 줄: | 38번째 줄: | ||
* [[데데킨트 합]] | * [[데데킨트 합]] | ||
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| + | ==수학용어번역== | ||
| + | * {{수학용어집|url=residue}} | ||
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| − | + | ==사전 형태의 자료== | |
| + | * http://en.wikipedia.org/wiki/residue_theorem | ||
| − | + | [[분류:복소함수론]] | |
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| − | + | == 노트 == | |
| − | + | ===말뭉치=== | |
| + | # Applying the Cauchy residue theorem.<ref name="ref_a923acc6">[https://math.libretexts.org/Bookshelves/Analysis/Book%3A_Complex_Variables_with_Applications_(Orloff)/09%3A_Residue_Theorem/9.05%3A_Cauchy_Residue_Theorem 9.5: Cauchy Residue Theorem]</ref> | ||
| + | # The integral over this curve can then be computed using the residue theorem.<ref name="ref_bc937111">[https://en.wikipedia.org/wiki/Residue_theorem Residue theorem]</ref> | ||
| + | # The new winding number allows to establish a generalized residue theorem which covers also the situation where singularities lie on the cycle.<ref name="ref_e674624a">[https://www.hindawi.com/journals/jmath/2019/6130464/ Non-Integer Valued Winding Numbers and a Generalized Residue Theorem]</ref> | ||
| + | # 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.<ref name="ref_e674624a" /> | ||
| + | # 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 .<ref name="ref_e674624a" /> | ||
| + | # Definition 2 of a generalized winding number turns out to be useful as it allows to generalize the residue theorem (see Theorem 8 below).<ref name="ref_e674624a" /> | ||
| + | # The Espil's theorem it's a short proof of the Cauchy's generalized residue theorem.<ref name="ref_7cb1076e">[https://zenodo.org/record/3359674 Espil short proof of generalized Cauchy's residue theorem]</ref> | ||
| + | # However, I decided to use the nuclear bomb of the integration arsenal, the Cauchy residue theorem of complex analysis.<ref name="ref_f1fcf6e1">[https://ekamperi.github.io/math/2020/12/15/cauchy-residue-theorem.html Computing improper integrals with Cauchy's residue theorem]</ref> | ||
| + | # In an upcoming topic we will formulate the Cauchy residue theorem.<ref name="ref_844a9d35">[https://www.cite-danper.com/blood-physiology-pqhvw/5c3232-cauchy-residue-theorem cauchy residue theorem]</ref> | ||
| + | # 1 Residue theorem problems 2 2 Zero Sum theorem for residues problems 76 3 Power series problems 157 Acknowledgement.<ref name="ref_844a9d35" /> | ||
| + | # The following result, Cauchy’s residue theorem, follows from our previous work on integrals.<ref name="ref_844a9d35" /> | ||
| + | # Using residue theorem to compute an integral.<ref name="ref_844a9d35" /> | ||
| + | ===소스=== | ||
| + | <references /> | ||
| − | == | + | == 메타데이터 == |
| − | + | ===위키데이터=== | |
| − | * | + | * ID : [https://www.wikidata.org/wiki/Q830513 Q830513] |
| − | + | ===Spacy 패턴 목록=== | |
| − | + | * [{'LOWER': 'residue'}, {'LEMMA': 'theorem'}] | |
| − | * [ | + | * [{'LOWER': 'cauchy'}, {'LOWER': 'residue'}, {'LEMMA': 'theorem'}] |
| − | * [ | + | * [{'LOWER': 'cauchy'}, {'LOWER': "'s"}, {'LOWER': 'residue'}, {'LEMMA': 'theorem'}] |
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2021년 2월 21일 (일) 19:43 기준 최신판
개요
- 복소함수론의 주요 정리 중 하나
응용
\[\sum_{k=1}^{\infty}\frac{1}{k^{4}-a^4}=\frac{1}{2a^4}-\frac{\pi \cot (\pi a)}{4 a^3}-\frac{\pi \coth (\pi a)}{4 a^3}\] \[\sum_{n=-\infty}^{\infty}\frac{1}{n^2+n+1}=\frac{2\pi \tanh \left(\frac{\sqrt{3} \pi }{2}\right)}{\sqrt{3}}\]
역사
메모
관련된 항목들
수학용어번역
- residue - 대한수학회 수학용어집
사전 형태의 자료
노트
말뭉치
- Applying the Cauchy residue theorem.[1]
- The integral over this curve can then be computed using the residue theorem.[2]
- The new winding number allows to establish a generalized residue theorem which covers also the situation where singularities lie on the cycle.[3]
- 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]
- 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]
- 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]
- The Espil's theorem it's a short proof of the Cauchy's generalized residue theorem.[4]
- However, I decided to use the nuclear bomb of the integration arsenal, the Cauchy residue theorem of complex analysis.[5]
- In an upcoming topic we will formulate the Cauchy residue theorem.[6]
- 1 Residue theorem problems 2 2 Zero Sum theorem for residues problems 76 3 Power series problems 157 Acknowledgement.[6]
- The following result, Cauchy’s residue theorem, follows from our previous work on integrals.[6]
- Using residue theorem to compute an integral.[6]
소스
메타데이터
위키데이터
- ID : Q830513
Spacy 패턴 목록
- [{'LOWER': 'residue'}, {'LEMMA': 'theorem'}]
- [{'LOWER': 'cauchy'}, {'LOWER': 'residue'}, {'LEMMA': 'theorem'}]
- [{'LOWER': 'cauchy'}, {'LOWER': "'s"}, {'LOWER': 'residue'}, {'LEMMA': 'theorem'}]