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 Complex Analysis Complex Analysis Math Forum

 November 8th, 2015, 02:47 AM #1 Newbie   Joined: Nov 2014 From: math Posts: 27 Thanks: 1 Cauchy Riemann equations, analyticity 1. Suppose $\displaystyle f(z)=\rho e^{i\phi}=u+iv$. Find the C-R equations for f in terms of $\displaystyle \rho,\phi,r,\theta$. I only know the C-R equations in polar form $\displaystyle ru_r=v_\theta,\quad -u_\theta=rv_r$. I can't imagine them in terms of 4 variables! Could someone give me a hint? Should I substitue $\displaystyle \rho,\phi$ into u, v? 2. Let $\displaystyle u(x,y)=\ln(x^2+y^2)$ be defined on $\displaystyle \Bbb C/\{0\}$. Find a harmonic conjugate $\displaystyle v(x,y)$ on some domain $\displaystyle \Omega \subset \Bbb C/\{0\}$ where $\displaystyle \Omega$ is of your choice. Also explain how you justify the analyticity of $\displaystyle u+iv$ on $\displaystyle \Omega$. I used the C-R equations and I get $\displaystyle v(x,y)=2\arctan \dfrac{y}{x}-2\arctan \dfrac{x}{y}+C$. How should I choose $\displaystyle \Omega$? And how can I justify the analyticity? Tags analyticity, cauchy, equations, riemann Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post jim198810 Complex Analysis 12 April 13th, 2015 07:53 AM Maximo Complex Analysis 1 February 7th, 2014 08:33 AM xboxlive89128 Complex Analysis 0 September 5th, 2009 01:24 PM maru1980 Number Theory 2 November 19th, 2007 12:47 AM hanahou Complex Analysis 1 October 28th, 2007 12:55 PM

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