For any two real numbers k and x1, the function G(z) k Ln(z x1) is analytic in the upper

Chapter 20, Problem 20

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For any two real numbers k and \(x_{1}\), the function \(G(z)=\) \(k \operatorname{Ln}\left(z-x_{1}\right)\) is analytic in the upper half-plane and therefore is a complex potential for a flow. The real number \(\(x_{1}\)\) is called a sink when \(k<0\) and a source for the flow when \(k>0\).

(a) Show that the streamlines are rays emanating from \(x_{1}\).

(b) Show that \(\mathbf{V}=\left(k /\left|z-x_{1}\right|^{2}\right)\left(z-x_{1}\right)\) and conclude that the flow is directed toward  \(x_{1}\) precisely when \(k<0\).