Construct a Complex Analytic Function

Construct a complex analytic function, starting from the values of its real and imaginary parts on the axis.

The real and imaginary parts u and v satisfy the Cauchy–Riemann equations.

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creqns = {D[u[x, y], x] == D[v[x, y], y], D[v[x, y], x] == -D[u[x, y], y]};

Prescribe the values of u and v on the axis.

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xvals = {u[x, 0] == x^3, v[x, 0] == 0};

Solve the Cauchy–Riemann equations.

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sol = DSolve[{creqns, xvals}, {u, v}, {x, y}]

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Verify that the solutions are harmonic functions.

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Laplacian[{u[x, y], v[x, y]} /. sol[[1]], {x, y}]

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Visualize the streamlines and equipotentials generated by the solution.

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ContourPlot[{u[x, y], v[x, y]} /. sol[[1]], {x, -5, 5}, {y, -5, 5}, ContourStyle -> {Red, Blue}]

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Construct a complex analytic function from the solution.

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f[x_, y_] = u[x, y] + I v[x, y] /. sol[[1]]

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This represents the function .

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(f[x, y] // Factor) /. {x + I y -> z}

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