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Complex analysis Jordans lemma
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This is a lecture from the middle of an undergraduate course on complex analysis. By this point in the course, the students have learned how to do various kinds of contour integrals with residue theory. In this lecture, Professor Strogatz shows how to finish the calculation of the integral of sin(x)/x, from -infinity to infinity, by relating it to the integral of exp(iz)/z on an indented contour in the complex plane. The lecture focuses on how to show the relevant integral over a large semicircle C_R in the upper half plane tends to zero as its radius R tends to infinity. They key step is the use of Jordan's lemma. This lemma yields an upper bound on the magnitude of the integral over C_R, and its proof is very clever. Prof. Strogatz gently walks the students through the proof and explains when Jordan's lemma is necessary and how it generalizes.","styleRuns":[{"startIndex":0,"length":857,"styleRunExtensions":{"styleRunColorMapExtension":{"colorMap":[{"key":"USER_INTERFACE_THEME_DARK","value":4294967295},{"key":"USER_INTERFACE_THEME_LIGHT","value":4279440147}]}},"fontFamilyName":"Roboto"}]},"headerRuns":[{"startIndex":0,"length":857,"headerMapping":"ATTRIBUTED_STRING_HEADER_MAPPING_UNSPECIFIED"}]}},{"itemSectionRenderer":{"contents":[{"videoMetadataCarouselViewModel":{"carouselTitles":[{"carouselTitleViewModel":{"title":"Live chat opnieuw 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