find the analytic function f(z)=u+iv,given u=a(1+cosx)

Answer 1

Generally since we are dealing with complex representations, the function, f(z) = u + iv is generally an equations of two variables (x,y) representing the complex plane. In the case provided, since y is not present in u, u(x,y) = u(x), or u(x,y) is constant in y.

Analytic functions require the Cauchy-Riemann conditions to be present which states...

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

∂u/∂x = -a * sin(x)

∂u/∂y = 0

Therefore we can now solve for v (at least to an arbitrary constant)

∂v/∂y = -a * sin(x) --> v(x,y) = -ay*sin(x) + C

∂v/∂x = -ay*cos(x) as per the definition above for the v(x,y)

Now we check to see if we can find ∂v/∂x = -∂u/∂y

-ay*cos(x) = 0 --> a = 0 for arbitrary x (only when x = nπ/2 for n ∈ integers would a ≠ 0)