{"id":4609,"date":"2021-08-03T10:13:07","date_gmt":"2021-08-03T04:43:07","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=4609"},"modified":"2025-12-22T10:10:20","modified_gmt":"2025-12-22T04:40:20","slug":"what-is-an-analytic-function","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-an-analytic-function\/","title":{"rendered":"What is an analytic function?"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><strong>Overview<\/strong>(analytic function)<\/h2>\n\n\n\n<p>A function&nbsp; is said to be <a href=\"https:\/\/en.wikipedia.org\/wiki\/Analytic_function\" target=\"_blank\" rel=\"noreferrer noopener\">analytic function<\/a> at a point<sub> <\/sub>z<sub>0<\/sub> if f is differentiable not only at z<sub>0<\/sub> but an every point of some neighborhoods at z<sub>0<\/sub>.<\/p>\n\n\n\n<p><strong>Note-<\/strong><\/p>\n\n\n\n<p>1. A point at which the function is not differentiable is called <strong>singular point.<\/strong><\/p>\n\n\n\n<p>2. A function which is analytic everywhere is called an <strong>entire function.<\/strong><\/p>\n\n\n\n<p>3. An entire function is always analytic, differentiable and continuous function.( converse is not true)<\/p>\n\n\n\n<p>4. Analytic function is always differentiable and continuous but converse is not true.<\/p>\n\n\n\n<p>5. A differentiable function is always continuous but converse is not true.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>The necessary condition for f(z) to be analytic<\/strong><\/h2>\n\n\n\n<p>f(z) = u + i(v) is to be analytic at all the points in a region R are-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/PZwNrLvM12f2EAG8fMZGfxYByExSy9xEUpsOe_LL3ivNDV4v7PrwUXB8jfwVn9KiFePvq3cLHqU9js8Fduik9Szj6y3De-kF0eA3seBqeaEw2T2_o8O0ak1G5OVnEkZA_YmLJorE\" alt=\"\"\/><\/figure>\n\n\n\n<p>Provided <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/UIPAfZ6OD_uA0pz78OAR1eAHoKeWPWsrqnxCOTfwOzSD6egYhWYROCW5pLkNWA_nLn09vor3ZY4t2B1mPc7i1x_Q281eSpYGqmvn4l8CK9iQT5daBFo4AqlYYBgqwkPktPFuKkTe\" width=\"85\" height=\"28\"> exists<\/p>\n\n\n\n<p>Equation (1) and (2) are known as Cauchy-Riemann equations.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>The sufficient condition for f(z) to be analytic<\/strong><\/h2>\n\n\n\n<p>f(z) = u + i(v) is to be analytic at all the points in a region R are-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/8qJ3OsN3bb07x9TndFVfVQbN3qPardn0IBfp1By5NABA9eOKxW6VdqJnEmTW5WzbPpNiMzerIdu9ZJQnX78EnpVS489l51LaeTxfQYPjRSYwu1Gv3M1YfAGe_TmOa9sO0OlW-1Ty\" alt=\"\"\/><\/figure>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/_sdxVkxlxgN96NSCnKsNszYGbV1GkXQx1__G7tjmV7GSGFGRFA3cRyc1QpxRR2iQJxqQ8UB8kadqG1J9hNH8oVfw-pr_KltnyUiKz3ATM7YHnrbcrDwstfUBW5BXziWMdtam397R\" width=\"90\" height=\"35\"> are continuous function of x and y in region R.<\/p>\n\n\n\n<p><strong>Important note-<\/strong><\/p>\n\n\n\n<p>1. If a function is analytic in a domain D, then u and v will satisfy Cauchy-Riemann conditions.<\/p>\n\n\n\n<p>2. C-R conditions are necessary but not sufficient for analytic function.<\/p>\n\n\n\n<p>3. C-R conditions are sufficient if the partial derivatives are continuous.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solved examples of analytic functions<\/h2>\n\n\n\n<p><strong>Example: If w = log z, then find dw\/dz. Also determine where w is non-analytic.<\/strong><\/p>\n\n\n\n<p>Sol. Here we have<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/G1VYfZwc6Nf1n4jsKQtpFCGpx241xZKA-4oMr9NI8s67Z6vYaryPhI0-47liPDU_mJf--APJUbAzCtnyEnYbIaJDYpdm4M61KdbcaV6fHwju9X53avMfFWWQ6OCHL2hld89x_NMa\" alt=\"\"\/><\/figure>\n\n\n\n<p>&nbsp;Therefore-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/oCf77Eqod76dAYcidVJivod0eVdzlRAkuXJrZgOkvXoolB79-5nkJA1WfJAvcDCkQYDZbtJe0DTlkXvFjTahJPaxKtoiotZJDuWvNiKPK7OVm6H2uRtzXtAbPckhJrjWMcIeYoDF\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/PHNp5B7lzJClFcerxt0xOMcvQkXGvGt7uEwHRc3kDZKybi_3R8aGBrAqaiqWLbHs8hDLIoIfnKNmyMxMyO8XijyZZSERCHjr8gGQGEklmJgoEIr_wmD9qAvwjNXjSUWIXgZb_BG_\" alt=\"\"\/><\/figure>\n\n\n\n<p>Again<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/T--c8pOswvhTpQWWKxJfpN9M8DacrsKQmiBZU7prhyDeAp3caiSvM1AMrU65Du8xm-CYMzNy-jP8b_ieO2_V5GnfLvFHyfQvG3cOboFpx8XHpdfZaHs0WuxJW6rYrcZ4O0ew6UrX\" alt=\"\"\/><\/figure>\n\n\n\n<p>Hence the C-R conditions are satisfied also the partial derivatives are continuous except at (0 , 0).&nbsp;<\/p>\n\n\n\n<p>So that w is analytic everywhere but not at z = 0<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/H5rEAfk0MMs-uuTZN1ze5QICucRH8IUxVZEg_9LWI1bZKKzL63qhSUVPUjJhnWMiXOs5fXR0pc7fKUo5nsWG_sdz_bDQru1JwScomD6eiLObjk4dPnt0BGfqltie9MhdszdUJGHL\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Example: Prove that the function&nbsp; is an analytical function.<\/strong><\/p>\n\n\n\n<p><strong>Sol:<\/strong><\/p>\n\n\n\n<p>Let<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/iJFmG87Ycu5Fn6aEcnNnDgacsdFXrtFg1wNiMf7ZxLGMDZtdYRL0xCbUF6pe4R-ZbsL1ZkwwfUpw5d2M1ObsK105QfGn_ueZrmIK3plWGyKqNb2l68JdjH7p31JsNb5f1uftDLeH\" alt=\"\"\/><\/figure>\n\n\n\n<p>Let<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/_xSY_Ucp9y_3lgDZy_9Rp2ALS5oSk64xP3CFnp4VNUXMN-6W9XYylzruN3TejlV4wV0FGZrEgSGJflOGkLdhcjC6CIHCM2CL6v-Xi0Z8G4h_E9GvXyA-RCZ2Fi-TzRvsayQEXZlF\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/15UgAfpPRmwLiWWzZdHfr-OGjoIG2XzKNT4CeweRuoGvb59XKze9iUqZYbtwKlzDVZyWrO5QvfdFhRGQQwYaFRyBinngnXUdaAxYBgVuOX_XTHE1UB521J3dqPs2HRRjvAkaCPao\" alt=\"\"\/><\/figure>\n\n\n\n<p>Hence C-R-Equation satisfied.<\/p>\n\n\n\n<p><strong>Example<\/strong>: <strong>Show that polar form of C-R equations are-<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/EM4n1QmpT9T_EL_Ap-l2pFoxrTfqIVqHnX1pmWPF2WEdT8kK97Vw-Z-LM4lLHe3sMOa-07sKSJTBrDwXSpiJrFyscevOEYl4xUz6fUIud1ZBlnPVJMOEyp51HwUoVXY_l9Q0iV2v\" alt=\"\"\/><\/figure>\n\n\n\n<p>Sol:<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/jQzXmoJKTLoQpv3BudT-aCNXtgDNWzVJqn5n7P1ZuPfXOrqd1158RzN6QKjOC26EMyw_1XgqmYM7JjmtM4NVURgPRNFuZeBaHXDpcvlw78oFaTNuFryFeFlW0QwnQysXOC2qhPiW\" alt=\"\"\/><\/figure>\n\n\n\n<p>U and v are expressed in terms of r and \u03b8.<\/p>\n\n\n\n<p>Differentiate it partially w.r.t. r and \u03b8, we get-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/2VcFm-4SKe3ij8jYlfQdxweOCa5XPENqxZKH6L3VDoFNfTbFY0l0-PuA73J2SQwd9y8fwreHIGurPryQHCHpBe8yT6S88mNkrZSfK6oDNdx0KDAnROuvghwCWoG0q2Ap4yNLjXVV\" alt=\"\"\/><\/figure>\n\n\n\n<p>By equating real and imaginary parts, we get-<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/qNAK0CkXKvfoaLvmaHq0uwFPKj7PzlqotCIBrLqzH97QpU1l_4aTlqoN4lSoSBZcUOgbR8CvnI1pQSRKxqU0sZ5kXwMnm6CfMYj7GNeC4M4ghXWEUeoLRp1HxDmUeyHBzMcy4dBR\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Int<\/strong><strong>erested in learning about similar topics? Here are a few hand-picked blogs for you!<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-a-linear-differential-equation\/\" target=\"_blank\" rel=\"noreferrer noopener\">What are linear differential equations?<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-partial-differentiation\/\" target=\"_blank\" rel=\"noreferrer noopener\">What is partial differentiation?<\/a> <a href=\"https:\/\/oyo88.one\/\">https:\/\/oyo88.one\/<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-curvature-and-radius-of-curvature\/\" target=\"_blank\" rel=\"noreferrer noopener\">curvature<\/a>. <a href=\"https:\/\/outbackacademy.org.au\/\">https:\/\/outbackacademy.org.au\/<\/a><\/li>\n<\/ul>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>A function is said to be analytic function at a point z0 if f is differentiable not only at z0 but an every point of some neighborhoods at z0.<\/p>\n","protected":false},"author":28,"featured_media":4794,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31],"tags":[],"class_list":["post-4609","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.3.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is an analytic function? 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