경로적분,contour_integral


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)

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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).

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thm

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Misc

같은 한국어, 다른 영어: 물리학의 경로적분,path_integral by Richard_Feynman. QM에 해당하는거라
[https]물리학백과: 경로 적분 Path integral - 뭔지만 대충 설명하고 수식전개 등 자세히 하지는 않음.
WpEn:Path_integral은 line integral, contour integral과의 혼동을 우려해 disambiguation page로 처리.
https://everything2.com/title/path integral


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