{"id":99,"date":"2005-10-28T14:49:51","date_gmt":"2005-10-28T14:49:51","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=99"},"modified":"2007-04-29T20:09:07","modified_gmt":"2007-04-29T20:09:07","slug":"cauchy-riemann","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=99","title":{"rendered":"Cauchy-Riemann"},"content":{"rendered":"

The Cauchy-Riemann equations are one of the first results one comes across in Complex Analysis. A poster on S.O.S. Mathematics Cyberboard<\/a> has pointed that that proofs like that at Cauchy-Riemann equations<\/a> tend to take it for granted that if f(x+iy)=u+iv is analytic then the partial derivatives of u and v exist. Thus the proof at Cauchy-Riemann equations<\/a> says<\/p>\n

\\displaystyle f^{\\prime}(z)=\\lim_{h\\rightarrow 0}{\\left[\\frac{u(x+h,y)-u(x,y)}{h}+i\\frac{v(x+h,y)-v(x,y)}{h}\\right]} and then deduces that \\displaystyle f^{\\prime}(z)=\\frac{\\partial u}{\\partial x} + i\\frac{\\partial v}{\\partial x}<\/p>\n

Looking at various textbooks this omission seems to the norm. Even Ahlfors Complex Analysis<\/a> says: We remark that the existence of the … partial derivatives … is implied by the existence of <\/em>f^{\\prime}(z)<\/p>\n

One excellent book A First Course in Complex Functions by G.J.O. Jameson<\/a> does give a proper proof of this result. It defines differentiability for u:A \\to \\mathbb{R}^2 (where A is a subset of \\mathbb{R}^2) at a point (a,b) in the interior of A if there exists real numbers \\lambda,\\mu such that, given \\epsilon&gt;0, there exists \\delta&gt;0 such that, for all real h,k with \\sqrt{h^2+k^2}&lt; \\delta, |u(a+h,b+k)-u(a,b)-(\\lambda h + \\mu k)|\\le\\epsilon \\sqrt{h^2+k^2}<\/p>\n

Putting k=0 shows that \\lambda = \\dfrac{\\partial u}{\\partial x}; similarly \\mu = \\dfrac{\\partial u}{\\partial y}<\/p>\n

If f^{\\prime}(a+ib)=\\lambda +i\\mu then, given \\epsilon&gt;0, there exists \\delta&gt;0 such that for all real h,k with |h+ik|&lt;\\delta<\/p>\n

|f((a+h)-i(b+k))-f(a+ib)-(\\lambda + i\\mu)(h+ik)|\\le \\epsilon|h+ik|<\/p>\n

and taking real parts<\/p>\n

|u(a+h,b+k)-u(a,b)-(\\lambda h - \\mu k)|\\le\\epsilon \\sqrt{h^2+k^2} from which it follows that \\dfrac{\\partial u}{\\partial x} and \\dfrac{\\partial u}{\\partial y} exist. Taking imaginary parts gives the other 2 partial derivatives.<\/p>\n","protected":false},"excerpt":{"rendered":"

The Cauchy-Riemann equations are one of the first results one comes across in Complex Analysis. A poster on S.O.S. Mathematics Cyberboard has pointed that that proofs like that at Cauchy-Riemann equations tend to take it for granted that if is analytic then the partial derivatives of and exist. Thus the proof at Cauchy-Riemann equations says […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-99","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/99","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=99"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/99\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=99"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=99"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=99"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}