# Wolfram Language™

## 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 CauchyRiemann equations.

In:= ```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.

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

Solve the CauchyRiemann equations.

In:= `sol = DSolve[{creqns, xvals}, {u, v}, {x, y}]`
Out= Verify that the solutions are harmonic functions.

In:= `Laplacian[{u[x, y], v[x, y]} /. sol[], {x, y}]`
Out= Visualize the streamlines and equipotentials generated by the solution.

In:= ```ContourPlot[{u[x, y], v[x, y]} /. sol[], {x, -5, 5}, {y, -5, 5}, ContourStyle -> {Red, Blue}]```
Out= Construct a complex analytic function from the solution.

In:= `f[x_, y_] = u[x, y] + I v[x, y] /. sol[]`
Out= This represents the function .

In:= `(f[x, y] // Factor) /. {x + I y -> z}`
Out= 