Prove that the function: \(u(x, y)=(x-1)^3-3 x y^2+3 y^2\) is harmonic and find its harmonic conjugate and the corresponding analytic function \(f(z)\) in terms of \(z \).
In this problem, we are tasked with proving that the function u(x,y)=(x-1)^(3)-3xy^(2)+3y^(2)u(x, y) = (x-1)^3 – 3xy^2 + 3y^2 is harmonic. Furthermore, we need to find its harmonic conjugate and express the corresponding analytic function f(z)f(z) in terms of zz.
Step 1: Verification of Harmonicity:
To begin, we need to determine whether the given function u(x,y)u(x, y) is harmonic. We calculate its partial derivatives with respect to xx and yy:
We notice that (del^(2)u)/(delx^(2))+(del^(2)u)/(dely^(2))=6(x-1)-6(x-1)=0\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 6(x-1) – 6(x-1) = 0. This confirms that uu is a harmonic function.
Step 2: Harmonic Conjugate and Analytic Function f(z)f(z):
Next, we aim to find the harmonic conjugate of uu and express the corresponding analytic function f(z)f(z) in terms of zz.
We recognize that u(x,y)u(x, y) is the real part of the analytic function f(z)f(z), where f(z)=u(x,y)+iv(x,y)f(z) = u(x, y) + iv(x, y).
We utilize the Cauchy-Riemann equations, which state that u_(x)=v_(y)u_x = v_y and u_(y)=-v_(x)u_y = -v_x, to determine the imaginary part v(x,y)v(x, y) of f(z)f(z):
Finally, we express the analytic function f(z)f(z) in terms of zz. We use the result that f(z)=(z-1)^(3)+Cf(z) = (z-1)^3 + C
Conclusion:
In conclusion, the given function u(x,y)u(x, y) is confirmed to be harmonic. We have found its harmonic conjugate, v(x,y)v(x, y), and expressed the corresponding analytic function f(z)f(z) as f(z)=(z-1)^(3)+Cf(z) = (z-1)^3 + C, where CC is an arbitrary constant.