Answer to Question #337427 in Complex Analysis for Genius

Question #337427

suppose f(z) =1/z. write f in the form f(z) = u(x,y) + iv(x,y), where z = x+iy and u and v are real-valued functions


1
Expert's answer
2022-05-06T05:40:33-0400

"z=x+iy", where "x,y\\in{\\mathbb{R}}". We receive: "\\frac{1}{z}=\\frac{1}{x+iy}=\\frac{x-iy}{(x+iy)(x-iy)}=\\frac{x-iy}{x^2+y^2}." From the latter expression we receive that "u=\\frac{x}{x^2+y^2}" and "v=\\frac{-y}{x^2+y^2}".

Answer: "f(z)=u(x,y)+iv(x,y)," where "z=x+iy", "x,y\\in{\\mathbb{R}}"; "u(x,y)=\\frac{x}{x^2+y^2}", "v(x,y)=\\frac{-y}{x^2+y^2}".


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