Question

In fluid mechanics the velocity potential ϕ and the stream function ψ are such that f(z)= ϕ + iψ where f(z) is an analytic function.

If ϕ= x²+4x−y²+2y, find f(z).

Answer 1

To find f(z), we need to find both the real part (ϕ) and the imaginary part (ψ). Given that ϕ = x² + 4x - y² + 2y, we can determine the real part of f(z) by taking the partial derivative of ϕ with respect to x and then multiplying it by i. The resulting function will be the imaginary part of f(z).

Step-by-step explanation:

To find the function f(z), we need to find both the real part (ϕ) and the imaginary part (ψ). Given that ϕ = x² + 4x - y² + 2y, we can determine the real part of f(z) by taking the partial derivative of ϕ with respect to x and then multiplying it by i. The resulting function will be the imaginary part of f(z).

Taking the partial derivative of ϕ with respect to x, we get d(ϕ)/dx = 2x + 4. Multiplying it by i, we obtain i(2x + 4) = 4i + 2ix. So the real part of f(z) is 4i + 2ix.

Finally, we can write f(z) as f(z) = ϕ + iψ. Substituting the expressions for the real and imaginary parts, we have f(z) = (x² + 4x - y² + 2y) + (4i + 2ix) = x² + 2ix + 4x - y² + 2y + 4i.