{"id":22291,"date":"2022-08-31T00:59:16","date_gmt":"2022-08-30T16:59:16","guid":{"rendered":"https:\/\/ocw.nycu.edu.tw\/?post_type=course_page&#038;p=22291"},"modified":"2026-03-23T10:32:42","modified_gmt":"2026-03-23T02:32:42","slug":"cv962_970606","status":"publish","type":"course_page","link":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e8%a4%87%e8%ae%8a%e5%87%bd%e6%95%b8%e8%ab%96-complex-analysis-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb-%e5%90%b3%e5%9f%b9%e5%85%83%e8%80%81%e5%b8%ab\/cv962_970606","title":{"rendered":"Section77 Integration Along a Branch Cut, Section78 Definite Integrals Involving Sines and Cosines"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"22291\" class=\"elementor elementor-22291\">\n\t\t\t\t\t\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-10 elementor-section elementor-top-section elementor-element elementor-element-5489826f elementor-section-boxed 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100%;-webkit-mask-repeat:var(--divider-pattern-repeat);mask-repeat:var(--divider-pattern-repeat);background-color:var(--divider-color);-webkit-mask-image:var(--divider-pattern-url);mask-image:var(--divider-pattern-url)}.elementor-widget-divider--no-spacing{--divider-pattern-size:auto}.elementor-widget-divider--bg-round{--divider-pattern-repeat:round}.rtl .elementor-widget-divider .elementor-divider__text{direction:rtl}.e-con-inner>.elementor-widget-divider,.e-con>.elementor-widget-divider{width:var(--container-widget-width,100%);--flex-grow:var(--container-widget-flex-grow)}<\/style>\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-61f83b1f elementor-widget elementor-widget-shortcode\" data-id=\"61f83b1f\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\">\n<table id=\"tablepress-34\" class=\"tablepress tablepress-id-34\">\n<thead>\n<tr class=\"row-1 odd\">\n\t<th class=\"column-1\">\u9031\u6b21<\/th><th class=\"column-2\">\u8ab2\u7a0b\u5167\u5bb9<\/th><th class=\"column-3\">\u8ab2\u7a0b\u5f71\u97f3<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-hover\">\n<tr class=\"row-2 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">\u7b2c\u4e00\u7ae0 Complex Numbers<br \/>\nSection1 Sums and Products <br \/>\nSection2 Basic Algebraic Properties Further Properties <br \/>\nSection3 Moduli <br \/>\nSection4 Complex Conjugates<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=20062\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-3 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section5 Polor Form<br \/>\nSection6 Exponential Form<br \/>\nSection7 Powers and Roots<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22130\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-4 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">\u7b2c\u4e00\u7ae0 Complex Numbers<br \/>\nSection8 Regions in the Complex Plane<br \/>\n\u7b2c\u4e8c\u7ae0 Analytic Functions<br \/>\nSection9 Functions of a Complex Variable<br \/>\nSection10 Mappings<br \/>\nSection11 Mappings by the Exponential Function<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22142\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-5 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section12 Limits<br \/>\nSection13 Theorems in Limits<br \/>\nSection14 Derivatives<br \/>\nSection15 Differentiation Formulas<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22150\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-6 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section16 Limits Involving the Points at Infinity<br \/>\nSection17 Continuity<br \/>\nSection18 Differentiatility<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22156\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-7 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section18 Differentiatility<br \/>\nSection19 Differentiatility Formulas <br \/>\nSection20 Cauchy-Riemann Equations <br \/>\nSection21 Reflectiotn Principle <br \/>\nSection22 C-R equation in Polar Form<br \/>\nSection23 Analytic Function<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22162\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-8 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section23 Analytic Functions <br \/>\nSection24 Harmonic Functions<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22168\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-9 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section25 Harmonic Functions-1<br \/>\nSection26 Uniquely Determined Analytic Functions<br \/>\nSection27 Refelction Principle<br \/>\n\u7b2c\u4e09\u7ae0 Elementary Functions<br \/>\nSection28 The Exponential Func<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22174\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-10 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section29 The Logarithmic Function <br \/>\nSection31 Branches and Derivatives of Logarithms<br \/>\nSection32 Complex Exponents Functions<br \/>\nSection33 Trigonometric Functions<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22180\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-11 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section34 Hyperbolic Functions<br \/>\nSection35 Inverse Trigonometric and Hyperbolic Functions<br \/>\nSection36 Derviatives of Functions<br \/>\nSection37 Integrals<br \/>\nSection38 Contours<br \/>\nSection39 Contours Integrals<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22186\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-12 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section40 Examples<br \/>\nSection41 Antiderivatives<br \/>\nSection42 Fundamental Theory of Integral<br \/>\nSection43 Examples<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22195\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-13 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section44 Cauchy-Goursat Theorem (VersionI)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22201\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-14 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section45 Cauchy-Goursat Theorem (VersionII) <br \/>\nSection46 Cauchy-Goursat Theorem (VersionIII)<br \/>\nSection47 Cauchy Integral Formula<br \/>\nSection48 Derivatives of Analytic Functions<br \/>\nSection49 Liouvilles Theorem<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22207\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-15 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section50 Maximum Modulus Principle<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22213\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-16 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section50 Maximum Modulus Principle<br \/>\n\u7b2c\u4e94\u7ae0 Series<br \/>\nSection51 Convergence of Sequences<br \/>\nSection52 Convergence of Series<br \/>\nSection53 Taylor Series<br \/>\nSection54 Examples<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22219\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-17 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section55 Laurent Series<br \/>\nSection56 Examples<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22225\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-18 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section56 Examples<br \/>\nSection57 Absolate amd Uniform Convergence of Power Series<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22231\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-19 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section58 Continuity of Sums of Power Series<br \/>\nSection59 Integration and Differentiation of Power Series<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22237\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-20 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section59 Integration and Differentiation of Power Series<br \/>\nSection60 Uniqueness of Series Representations<br \/>\nSection61 Multiplication and Division of Power Series<br \/>\n\u7b2c\u516d\u7ae0 Residues and Poles<br \/>\nSection62 Residues<br \/>\nSection63 Cauchy<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22243\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-21 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section61 Multiplication and Division of Power Series<br \/>\n\u7b2c\u516d\u7ae0 Residues and Poles<br \/>\nSection62 Residues<br \/>\nSection63 Cauchys Residue Theorem<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22249\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-22 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section63 Cauchys Residue Theorem<br \/>\nSection64 Using a single Residue<br \/>\nSection65 The Three Types of Isolated Singular Points<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22255\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-23 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section66 Residues at Poles<br \/>\nSection67 Examples<br \/>\nSection68 Zeros of Analytic Functions<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22261\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-24 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section69 Zeros and Poles<br \/>\nSection70 Behavior of f Near Isolated Singular Points<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22267\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-25 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section70 Behavior of f Near Isolated Singular Points<br \/>\n\u7b2c\u4e03\u7ae0 Residues and Poles<br \/>\nSection71 Evaluation of Improper Integrals<br \/>\nSection72 Example<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22279\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-26 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section73 Improper Integrals from Fourier Analysis<br \/>\nSection74 Jordans Lemma<br \/>\nSection75 Indented PAths<br \/>\nSection76 An Indentation Around a Branch Point<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22285\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-27 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section77 Integration Along a Branch Cut<br \/>\nSection78 Definite Integrals Involving Sines and Cosines<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22291\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-28 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Section79 Argument Principle<br \/>\nSection80 Rouches Theorem<br \/>\nSection81 Inverse Laplace Transforms<br \/>\nSection82 Examples<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22297\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-29 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">\u7b2c\u516b\u7ae0 Mapping by Elementary Functions<br \/>\nSection86 Linear Fractional Transformations<br \/>\n\u7b2c\u4e5d\u7ae0Conformal Mapping<br \/>\nSection94 Preservation of Angles<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=22303\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-34 from cache --><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-3 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-3839e8c9\" data-id=\"3839e8c9\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7d945485 elementor-widget-divider--view-line_text elementor-widget-divider--element-align-center elementor-widget elementor-widget-divider\" data-id=\"7d945485\" data-element_type=\"widget\" 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