https://haifei-home.pages.dev/en/tags/mathematical-analysis/Recent content in Mathematical Analysis on Haifei's HomeHugoenhfwang132@gmail.com (hfwang132)hfwang132@gmail.com (hfwang132)This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.Wed, 09 Dec 2020 00:22:44 +0800https://haifei-home.pages.dev/en/post_20201209_%E5%88%AB%E5%86%8D%E8%A2%AB%E6%95%B0%E5%AD%A6%E7%B3%BB%E5%90%8C%E5%AD%A6%E9%84%99%E8%A7%86%E4%BA%86-%E7%89%A9%E7%90%86%E4%BA%BA%E5%81%9A%E6%8D%A2%E5%BA%8F%E6%93%8D%E4%BD%9C%E6%97%B6%E9%9C%80%E8%A6%81%E7%9F%A5%E9%81%93%E7%9A%84%E4%B8%80%E8%87%B4%E6%94%B6%E6%95%9B/Wed, 09 Dec 2020 00:22:44 +0800hfwang132@gmail.com (hfwang132)https://haifei-home.pages.dev/en/post_20201209_%E5%88%AB%E5%86%8D%E8%A2%AB%E6%95%B0%E5%AD%A6%E7%B3%BB%E5%90%8C%E5%AD%A6%E9%84%99%E8%A7%86%E4%BA%86-%E7%89%A9%E7%90%86%E4%BA%BA%E5%81%9A%E6%8D%A2%E5%BA%8F%E6%93%8D%E4%BD%9C%E6%97%B6%E9%9C%80%E8%A6%81%E7%9F%A5%E9%81%93%E7%9A%84%E4%B8%80%E8%87%B4%E6%94%B6%E6%95%9B/<h2 id="preface">Preface</h2> <p>Whenever physicists perform operations such as differentiating under the integral sign or interchanging the order of integration, the mathematicians watching can no longer sit still:</p> <p>“Does your improper integral/series converge uniformly?”</p> <p>The physicist replies:</p> <p>“What is uniform convergence? We have always done it this way.”</p> <p>Or:</p> <p>“Assume that this function has sufficiently nice properties.”</p> <p>Or, even more outrageously:</p> <p>“Assume that this function looks rather pretty.”</p> <figure class="post-figure" style="--post-figure-width: 80%;"><img src="https://haifei-home.pages.dev/post_20201209_%E5%88%AB%E5%86%8D%E8%A2%AB%E6%95%B0%E5%AD%A6%E7%B3%BB%E5%90%8C%E5%AD%A6%E9%84%99%E8%A7%86%E4%BA%86-%E7%89%A9%E7%90%86%E4%BA%BA%E5%81%9A%E6%8D%A2%E5%BA%8F%E6%93%8D%E4%BD%9C%E6%97%B6%E9%9C%80%E8%A6%81%E7%9F%A5%E9%81%93%E7%9A%84%E4%B8%80%E8%87%B4%E6%94%B6%E6%95%9B/images/v2-6c22ef6c5999b623aa4d9488888fba44_r.jpg" loading="lazy"/></figure> <p>looks good</p> <p>&hellip;</p>