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Verify by direct substitution that the functions rn cos(nθ) etc

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Verify by direct substitution that the functions rn cos(nθ) etc. shown in the text satisfy the Laplace equation in polar coordinates. Calculate the streamfunction for the above-mentioned potential function and construct a contour-plot of streamlines for n = 1/2, 1, and 3/2. MATLAB or MATHEMATICA may be used to make these contour-plots. Also construct a contour-plot of equipotential lines for any one of the above cases and show that these lines are perpendicular to the streamlines. Answer for streamfunction: ψ = rn sin(nθ). A constant can be added to the streamfunction without altering the results for the flow field. Hence verify that the complex function F(z) = zn is a solution to a potential problem where the real part becomes the velocity potential while the imaginary part represents the streamfunction.