"유수 정리 (residue theorem)"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
(→‎메타데이터: 새 문단)
52번째 줄: 52번째 줄:
  
 
[[분류:복소함수론]]
 
[[분류:복소함수론]]
−
 
−
== 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 ==
 
== 메타데이터 ==

2021년 2월 21일 (일) 20:43 판

개요


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

메타데이터

위키데이터

Spacy 패턴 목록

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

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]

소스

  1. ↑ 9.5: Cauchy Residue Theorem
  2. ↑ Residue theorem
  3. ↑ 3.0 3.1 3.2 3.3 Non-Integer Valued Winding Numbers and a Generalized Residue Theorem
  4. ↑ Espil short proof of generalized Cauchy's residue theorem
  5. ↑ Computing improper integrals with Cauchy's residue theorem
  6. ↑ 6.0 6.1 6.2 6.3 cauchy residue theorem

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'residue'}, {'LEMMA': 'theorem'}]
  • [{'LOWER': 'cauchy'}, {'LOWER': 'residue'}, {'LEMMA': 'theorem'}]
  • [{'LOWER': 'cauchy'}, {'LOWER': "'s"}, {'LOWER': 'residue'}, {'LEMMA': 'theorem'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=유수_정리_(residue_theorem)&oldid=52600"