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Magazine Problem 5
The given analytic function \(f(z)=u+i v\) defines two families of level curves \(u(x, y)=c_{1}\) and \(v(x, y)=c_{2} .\) First use implicit differentiation to compute \(d y / d x\) for each family and then verify that the families are orthogonal. \(f(z)=x-2 x^{2}+2 y^{2}+i(y-4 x y)\)
Problem 6
Verify that the given function \(u\) is harmonic in an appropriate domain \(D\). \(u(x, y)=\cos x \cosh y\)
Problem 6
The given analytic function \(f(z)=u+i v\) defines two families of level curves \(u(x, y)=c_{1}\) and \(v(x, y)=c_{2} .\) First use implicit differentiation to compute \(d y / d x\) for each family and then verify that the families are orthogonal. \(f(z)=x^{3}-3 x y^{2}+i\left(3 x^{2} y-y^{3}\right)\)
Problem 6
Show that the given function is not analytic at any point. \(f(z)=\bar{z}^{2}\)
Problem 7
Verify that the given function \(u\) is harmonic in an appropriate domain \(D\). \(u(x, y)=e^{x}(x \cos y-y \sin y)\)
Problem 7
The given analytic function \(f(z)=u+i v\) defines two families of level curves \(u(x, y)=c_{1}\) and \(v(x, y)=c_{2} .\) First use implicit differentiation to compute \(d y / d x\) for each family and then verify that the families are orthogonal. \(f(z)=e^{-x} \cos y-i e^{-x} \sin y\)
Problem 7
Show that the given function is not analytic at any point. \(f(z)=x^{2}+y^{2}\)
Problem 8
Show that the given function is not analytic at any point. \(f(z)=\frac{x}{x^{2}+y^{2}}+i \frac{y}{x^{2}+y^{2}}\)
Problem 8
The given analytic function \(f(z)=u+i v\) defines two families of level curves \(u(x, y)=c_{1}\) and \(v(x, y)=c_{2} .\) First use implicit differentiation to compute \(d y / d x\) for each family and then verify that the families are orthogonal. \(f(z)=x+\frac{x}{x^{2}+y^{2}}+i\left(y-\frac{y}{x^{2}+y^{2}}\right)\)
Problem 8
Verify that the given function \(u\) is harmonic in an appropriate domain \(D\). \(u(x, y)=-e^{-x} \sin y\)
