#noindex
[[복소평면,complex_plane]]에서의 [[선적분,line_integral]].[* [[WpEn:Contour_integration]] 맨 위 "This article is about the line integral in the complex plane."]
[[경로,contour]] 위에서 [[복소함수,complex_function]]의 적분.[* 수학백과 첫줄]

----
[[곡선,curve]]
폐곡선 closed curve's orientation:
||positive  ||ccw ||
||negative  ||cw ||



Def.
Let $f$ be defined at points of a smooth curve $C$ defined by
 $x=x(t),\,y=y(t),\;a\le t\le b.$
The '''contour integral''' of $f$ along $C$ is:
 $\int_C f(z)dz=\lim_{||P||\to0}\sum_{k=1}^{n} f(z_k^*) \Delta z_k$
(Zill AEM Def 18.1.1)

Thm. Evaluation of a Contour Integral
If $f$ is continuous on a smooth curve $C$ given by
 $z(t)=x(t)+iy(t),\;a\le t\le b,$
then:
 $\int_C f(z)dz=\int_a^b f(z(t))z'(t)dt$
(Zill AEM Thm 18.1.1)

= 성질 properties =
다음은 익숙
 * ∫,,C,, kf(z)dz= k∫,,C,, f(z)dz
 * ∫,,C,, (f(z) + g(z))dz = ∫,,C,, f(z)dz + ∫,,C,, g(z)dz
 * ∫,,C,, f(z)dz = ∫,,C₁,, f(z)dz + ∫,,C₂,, f(z)dz
 여기서 C는 매끄러운 곡선 C₁, C₂의 union
다음을 기억
 $\int_{-C}f(z)dz=-\int_C f(z)dz$ 여기서 -C는 반대 방향(orientation)을 의미(denote).

= thm =
[[코시-구르사_정리,Cauchy-Goursat_theorem]]
[[유수정리,residue_theorem]]

= Misc =
같은 한국어, 다른 영어: 물리학의 [[경로적분,path_integral]] by Richard_Feynman. QM에 해당하는거라
[[https://terms.naver.com/entry.naver?docId=5937873&cid=60217&categoryId=60217 물리학백과: 경로 적분 Path integral]] - 뭔지만 대충 설명하고 수식전개 등 자세히 하지는 않음.
[[WpEn:Path_integral]]은 line integral, contour integral과의 혼동을 우려해 disambiguation page로 처리.
https://everything2.com/title/path+integral

----
Twins:
[[https://terms.naver.com/entry.naver?docId=5668872&ref=y&cid=60207&categoryId=60207 수학백과: 경로적분법]]
[[WpKo:경로적분법]]
https://planetmath.org/contourintegral
https://mathworld.wolfram.com/ContourIntegral.html

Up:
[[경로,contour]]
[[적분,integration]]
[[복소해석,complex_analysis]]