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<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="ru"><front><journal-meta><journal-id journal-id-type="publisher-id">msi</journal-id><journal-title-group><journal-title xml:lang="ru">Современная наука и инновации</journal-title><trans-title-group xml:lang="en"><trans-title>Modern Science and Innovations</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">2307-910X</issn><publisher><publisher-name>North-Caucasus Federal University</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.33236/2307-910X-2020-2-30-102-104</article-id><article-id custom-type="elpub" pub-id-type="custom">msi-212</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>КРАТКИЕ СООБЩЕНИЯ</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="en"><subject>SHORT REPORTS</subject></subj-group></article-categories><title-group><article-title>СВОЙСТВА ОРТОГОНАЛЬНОСТИ И ПОЛНОТЫ СИСТЕМЫ СФЕРИЧЕСКИХ ФУНКЦИЙ</article-title><trans-title-group xml:lang="en"><trans-title>THE PROPERTIES OF ORTHOGONALITY AND COMPLETENESS OF THE SYSTEM OF SURFACE HARMONICS</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Абакумова</surname><given-names>С. И.</given-names></name><name name-style="western" xml:lang="en"><surname>Abakumova</surname><given-names>S. I.</given-names></name></name-alternatives><email xlink:type="simple">svetaabaku@rambler.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Ботвинева</surname><given-names>Н. Ю.</given-names></name><name name-style="western" xml:lang="en"><surname>Botvineva</surname><given-names>N. Y.</given-names></name></name-alternatives><email xlink:type="simple">noemail@neicon.ru</email><xref ref-type="aff" rid="aff-2"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Гулынина</surname><given-names>Е. В.</given-names></name><name name-style="western" xml:lang="en"><surname>Gulynina</surname><given-names>E. V.</given-names></name></name-alternatives><email xlink:type="simple">noemail@neicon.ru</email><xref ref-type="aff" rid="aff-2"/></contrib></contrib-group><aff-alternatives id="aff-1"><aff xml:lang="ru"><institution>Северо-Кавказский Федеральный Университет</institution><country>Россия</country></aff><aff xml:lang="en"><institution>North-Caucasus Federal University</institution><country>Russian Federation</country></aff></aff-alternatives><aff-alternatives id="aff-2"><aff xml:lang="ru"><institution>Ставропольский государственный педагогический институт</institution><country>Россия</country></aff><aff xml:lang="en"><institution>Stavropol State Pedagogical Institute</institution><country>Russian Federation</country></aff></aff-alternatives><pub-date pub-type="collection"><year>2020</year></pub-date><pub-date pub-type="epub"><day>04</day><month>08</month><year>2022</year></pub-date><volume>0</volume><issue>2</issue><fpage>111</fpage><lpage>113</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Абакумова С.И., Ботвинева Н.Ю., Гулынина Е.В., 2022</copyright-statement><copyright-year>2022</copyright-year><copyright-holder xml:lang="ru">Абакумова С.И., Ботвинева Н.Ю., Гулынина Е.В.</copyright-holder><copyright-holder xml:lang="en">Abakumova S.I., Botvineva N.Y., Gulynina E.V.</copyright-holder><license xml:lang="ru" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>Данная работа распространяется под лицензией Creative Commons Attribution 4.0.</license-p></license><license xml:lang="en" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>This work is licensed under a Creative Commons Attribution 4.0 License.</license-p></license></permissions><self-uri xlink:href="https://msi.elpub.ru/jour/article/view/212">https://msi.elpub.ru/jour/article/view/212</self-uri><abstract><p>Сферические функции представляют собой угловую часть семейства ортогональных решений уравнения Лапласа, записанную в сферических координатах. Использование этих функций достаточно разнообразно, они имеют большое значение в теории дифференциальных уравнений в частных производных и теоретической физике.</p></abstract><trans-abstract xml:lang="en"><p>Surface harmonics represent the angular part of the family of orthogonal solutions of the Laplace equation, written in spherical coordinates. The use of these functions is quite diverse, they are of great importance in the theory of differential equations in partial derivatives and theoretical physics.</p></trans-abstract><kwd-group xml:lang="ru"><kwd>сферические функции</kwd><kwd>ортогональность и полнота сферических функций</kwd><kwd>сферическая гармоника</kwd></kwd-group><kwd-group xml:lang="en"><kwd>surface harmonics</kwd><kwd>orthogonality and completeness of surface harmonics</kwd></kwd-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">Араманович И.Г., Левин. В.И. 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