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November 8th, 2015, 02:47 AM   #1
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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?
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