Editing Contour integration (section)

{{short description|Method of evaluating certain integrals along paths in the complex plane}}
{{About|the line integral in the complex plane|the general line integral|Line integral}}
{{Calculus |Integral}}

In the mathematical field of [[complex analysis]], '''contour integration''' is a method of evaluating certain [[integral]]s along paths in the [[complex plane]].<ref name=Stalker>{{Cite book|title=Complex Analysis: Fundamentals of the Classical Theory of Functions |first=John |last=Stalker |page=77 | url=https://books.google.com/books?id=yl3GIXd3dFIC&q=%22calculus+of+residues%22&pg=PP12|isbn=0-8176-4038-X |publisher=Springer |year=1998}}</ref><ref name=Bak>{{Cite book|title=Complex Analysis |first1=Joseph |last1=Bak |first2=Donald J. |last2=Newman |chapter=Chapters 11 & 12 |pages=130–156 |chapter-url=https://books.google.com/books?id=JX2YSgfZwbYC&q=%22contour+integral%22&pg=PA130 |isbn=0-387-94756-6 |year=1997 |publisher=Springer}}</ref><ref name=Krantz>{{Cite book|title=Handbook of Complex Variables |first=Steven George |last=Krantz |chapter= Chapter 2 |chapter-url=https://books.google.com/books?id=aYU2AdF_0dIC&q=Calculus++Residues+inauthor:krantz&pg=PT13 |isbn=0-8176-4011-8 |year=1999 |publisher=Springer }}</ref>

Contour integration is closely related to the [[Residue theorem|calculus of residues]],<ref name=Mitrinovic1>{{Cite book|title=The Cauchy Method of Residues: Theory and Applications |url=https://books.google.com/books?id=-suKhxfPH5AC&q=%22calculus+of+residues%22
|first1=Dragoslav S. |last1=Mitrinović |first2=Jovan D. |last2=Kečkić |isbn=90-277-1623-4 |year=1984 |publisher=Springer |chapter=Chapter 2 }}</ref> a method of [[complex analysis]].

One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. It also has various applications in physics.<ref name=Mitrinovic2>{{Cite book|chapter=Chapter 5 |title=The Cauchy Method of Residues: Theory and Applications|url=https://books.google.com/books?id=-suKhxfPH5AC&q=%22calculus+of+residues%22 |first1=Dragoslav S. |last1=Mitrinović |first2=Jovan D. |last2=Kečkić |year=1984 |publisher=Springer |isbn=90-277-1623-4 }}</ref>

Contour integration methods include:
* direct integration of a [[complex number|complex]]-valued function along a curve in the complex plane
* application of the [[Cauchy integral formula]]
* application of the [[residue theorem]]
One method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums.