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Approximate expressions are developed to estimate the orders and Mathieu characteristic numbers for the root, finding algorithms. The algorithms, with minor modifications, were used for computing Mathieu coefficients of general order. The algorithms can deal with a large range of Mathieu characteristic number\n            <jats:italic>c<\/jats:italic>\n            , real and complex order \u03bd, and parameter\n            <jats:italic>h<\/jats:italic>\n            .\n          <\/jats:p>","DOI":"10.1145\/1163641.1163646","type":"journal-article","created":{"date-parts":[[2006,10,18]],"date-time":"2006-10-18T18:11:32Z","timestamp":1161195092000},"page":"472-484","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":6,"title":["Algorithm 855"],"prefix":"10.1145","volume":"32","author":[{"given":"Fayez A.","family":"Alhargan","sequence":"first","affiliation":[{"name":"Computer and Electronics Research Institute, Riyadh, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2006,9]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1137\/1038040"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/358407.358422"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/358407.358420"},{"key":"e_1_2_2_4_1","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1049\/ip-map:20030504","article-title":"Cut-Off wavenumbers of sectoral elliptic resonators","volume":"150","author":"Alhargan F.","year":"2003","journal-title":"IEE Proc-Microw. 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Palermo (2) 15 51--97.","DOI":"10.1007\/BF02849408"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.1145\/363156.363176"},{"key":"e_1_2_2_10_1","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/BF01535268","article-title":"On the computation of Mathieu functions","volume":"7","author":"Delft Numerical Analysis Group","year":"1973","journal-title":"J. Eng. Math."},{"volume-title":"Proceedings of the Antennas and Propagation Society International Symposium, 318--321","author":"Do-Nhat T.","key":"e_1_2_2_11_1"},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-00-01227-8"},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1145\/355815.355824"},{"key":"e_1_2_2_14_1","unstructured":"McLachlan N. 1947. Theory and Applications of Mathieu Functions. Oxford University Press Oxford UK.  McLachlan N. 1947. Theory and Applications of Mathieu Functions. Oxford University Press Oxford UK."},{"key":"e_1_2_2_15_1","doi-asserted-by":"crossref","DOI":"10.1109\/TMTT.1980.1130060","article-title":"Mathieu functions of integral orders and real arguments","author":"Rengarajan S.","year":"1980","journal-title":"IEEE Trans. Microwave Theory Tech. MTT-28, 276--277."},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/155743.155847"},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/155743.155796"},{"key":"e_1_2_2_18_1","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1090\/S0025-5718-1962-0135739-3","article-title":"Characteristic exponents of Mathieu functions","volume":"16","author":"Tamir T.","year":"1962","journal-title":"Math. Comput."},{"key":"e_1_2_2_19_1","article-title":"Computer program descripation","author":"Toyama N.","year":"1984","journal-title":"IEEE Trans. Antennas Propag. 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