[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572660-105":53,"doc-detail-572660-en":120},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":113,"head_meta":115,"extra_data":117,"updated_unix":119},105,"en","contour-integration-cheat-sheet-residue-theorem-contours","Contour Integration Cheat Sheet - Residue Theorem & Contours","","Concise study notes for contour integration centered on residue computation and residue-based evaluation of complex integrals. Covers the definition of the residue via the Laurent expansion, states the Residue Theorem for piecewise smooth counterclockwise boundaries, and lists practical strategies for finding residues for partial fractions, simple poles, and double poles, plus a zero/pole residue formula. Includes a structured comparison of common contour choices (semicircle, unit circle, keyhole, and small indented arcs) with typical limits and integral-transformation guidance, followed by core references.",{"@graph":63,"@context":112},[64,80,95],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/contour-integration-cheat-sheet-residue-theorem-contours/572660/",4,{"url":78,"name":59,"@type":81,"author":82,"headline":59,"publisher":85,"fileFormat":88,"inLanguage":57,"description":61,"dateModified":89,"datePublished":89,"encodingFormat":88,"isAccessibleForFree":90,"interactionStatistic":91},"DigitalDocument",{"name":83,"@type":84},"Dusta123","Person",{"url":68,"name":86,"@type":87},"DocShare","Organization","application/pdf","2026-10-13",true,{"@type":92,"interactionType":93,"userInteractionCount":4},"InteractionCounter",{"@type":94},"ViewAction",{"@type":96,"mainEntity":97},"FAQPage",[98,104,108],{"name":99,"@type":100,"acceptedAnswer":101},"What is the residue of a function at an isolated singularity?","Question",{"text":102,"@type":103},"The residue Res[f; z0] is the coefficient a_{-1} of (z - z0)^{-1} in the Laurent expansion of f around z0.","Answer",{"name":105,"@type":100,"acceptedAnswer":106},"How does the Residue Theorem relate a contour integral to residues?",{"text":107,"@type":103},"For a bounded region D with a counterclockwise piecewise smooth boundary, the integral of f over @D equals 2πi times the sum of residues at singularities inside D.",{"name":109,"@type":100,"acceptedAnswer":110},"What strategies are given for computing residues in practice?",{"text":111,"@type":103},"The cheat sheet lists partial fraction decomposition, limits for simple poles, derivative-based limits for double poles, and a formula for the residue of a product when one factor has a simple zero.","https://schema.org",{"og:url":78,"og:type":114,"og:title":59,"og:site_name":86,"og:description":61},"article",{"robots":116,"canonical":78},"index,follow",{"doc_id":118,"site_id":56},572660,1791928754,{"code":4,"msg":5,"data":121},{"doc_id":118,"user_id":122,"nickname":83,"user_avatar":123,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":124,"file_id":125,"file_url":126,"file_type":127,"file_size":128,"view_count":4,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":79,"language":129,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":130,"faqs":131,"seo_title":132,"seo_description":61,"update_tm":119,"read_time":73},962093541925,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","Contour Integration Cheat Sheet  \nNatalie Frank  \nJuly 28, 2021  \nDe􀀌nition 1 (Residue) Assume that f has an isolated singularity at z0 . The residue of f atz0 , denoted Res[f; z0 ] is the coe􀀎cient a􀀀1 of (z 􀀀 z0 )􀀀1 in the Laurent expansion of f at z0 :  \n1  \nf (z) = X an (z 􀀀 z0 )n ; 0 \u003C j z 􀀀 z0 j \u003C 􀀚  \nn= 􀀀1  \nTheorem 1 (The Residue Theorem) Let D be a bounded set with a piecewise smooth boundary oriented in the counterclockwise direction. Assume that f is analytic on int D except at singularities z1 ; z2 ; : : : zm 2 int D . Then  \nm  \nZ@D f (z)dz = 2􀀙i Xi=1 Res[f(z); zj]  \nStrategies for 􀀌nding residues 2-4 on this list straight from in 7 . 1 of Complex Analysis by Gamelin.  \n1. Partial Fractions decomposition  \n2. If f(z) has a simple pole at z0 , then  \nRes[f(z); z0 ] = lim (z 􀀀 z0 )f (z)  \nz!z 0  \n3. If f(z) has a double pole at z0 , then  \n d   \nRes[f(z); z0 ] = lim ((z 􀀀 z0 )2 f (z))  \nz!z 0 dz  \n4. If f; g are analytic and g has a simple zero at z0 , then  \nRes[ ~~ ~~fg((zz)) ; z0 ] = fg0~~ ~~((zz00))  \nExercise 1 Prove 2-4 above.  \n\n| Contour | Uses |\n| --- | --- |\n| \u003Cbr>Semicircle Contour\u003Cbr>limits: R ! 1 | ❼ integral from 􀀀1 to 1\u003Cbr>❼ P (x)=Q (x) for Q; P polynomials with deg Q 􀀕 deg P + 2\u003Cbr>❼ P (x)=Q (x)R(sin(x) ; cos(x)) for P; Q; R polynomials with deg Q 􀀕 deg P + 1\u003Cbr>{ If in the previous bullet deg Q = deg P + 1, use Jordan's Lemma:\u003Cbr>Lemma 1 (Jordan's Lemma) Let 􀀀 R bethe semicircle of radius R in the upper half plane. Then\u003Cbr>Z􀀀 R jeiz jj dz j \u003C 􀀙 |\n| \u003Cbr>Unit Circle Contour | ❼ integral from 0 to 2􀀙\u003Cbr>❼ for rational functions of sin 􀀒; cos 􀀒\u003Cbr>❼ Goal: turn into an integral of rational functions around unit circle. Use the substitutions sin(z) = (eiz 􀀀 e􀀀iz )=2i; cos(z) = (eiz + e􀀀iz )=2 |\n| \u003Cbr>Keyhole Contour\u003Cbr>limits: 􀀏 ! 0; R ! 1, integrating along `1􀀏;R ; `2􀀏;R is implicitly a limit as well | ❼ integral from 0 to in􀀌nity, denominator a polynomial in P (x)\u003Cbr>❼ for integrand with a branch cut, typically za or log (z)\u003Cbr>❼ the integrand will assume di􀀋erent values on either side of the branch\u003Cbr>❼ In order for the integrand to have the right form on either side of the branch, irreducible factors of P should be (x􀀀a) for real a. One may 􀀌rst need to perform a change of variable if this is not the case |\n\n\n| Contour | Uses |\n| --- | --- |\n| \u003Cbr>small arc angle 􀀋, also called a\u003Cbr>\\contour indented at z0 \"\u003Cbr>limits: 􀀏 ! 0 | ❼ When integrating through a singularity on the real line and that singularity is a simple pole\u003Cbr>Theorem 2 (Fractional Residue Theorem\u003Cbr>Let z0 be a simple pole of f and let C􀀏 be the\u003Cbr>arc of angle 􀀋 (in radians) of radius 􀀏 . Then l􀀏im!0 ZC􀀏 f (z)dz = 􀀋i Res[f(z); z0 ) |\n\n)  \nReferences: This cheat-sheet summarizes 7 .1-7.5 and 7 .7 of Complex Analysis by Gamelin. All theorem statements are from this text as well. Look in the textbook for proofs, worked examples, and more information.","cbCait3cfrf9tYDr","https://ap.wps.com/l/cbCait3cfrf9tYDr","pdf",15460696,"English","# Definitions and Core Results\n## Residue and Residue Theorem\n## Strategies for Finding Residues\n## Fractional Residue Theorem\n# Common Contours and Uses\n## Semicircle Contour\n## Unit Circle Contour\n## Keyhole Contour\n## Small Arc / Indented Contour","[{\"question\":\"What is the residue of a function at an isolated singularity?\",\"answer\":\"The residue Res[f; z0] is the coefficient a_{-1} of (z - z0)^{-1} in the Laurent expansion of f around z0.\"},{\"question\":\"How does the Residue Theorem relate a contour integral to residues?\",\"answer\":\"For a bounded region D with a counterclockwise piecewise smooth boundary, the integral of f over @D equals 2πi times the sum of residues at singularities inside D.\"},{\"question\":\"What strategies are given for computing residues in practice?\",\"answer\":\"The cheat sheet lists partial fraction decomposition, limits for simple poles, derivative-based limits for double poles, and a formula for the residue of a product when one factor has a simple zero.\"}]","Contour Integration Cheat Sheet - Residue Theorem & Contours | PDF"]