Question Analysis:
We are given an analytic function f(z) = u + iv, where v = 2xy. We need to determine the correct option that represents the function.

Given:
v = 2xy

Solution:
Step 1: Analyzing the given function
The given function is f(z) = u + iv, where v = 2xy. Let's break down the function into its real and imaginary parts:
u = Re(f(z))
v = Im(f(z))

Step 2: Calculating the partial derivatives
To find the real and imaginary parts of the function, we need to calculate the partial derivatives of u and v with respect to x and y.

Partial derivative of u with respect to x:
∂u/∂x = ∂/∂x (u + iv) = ∂u/∂x + i∂v/∂x
Since v = 2xy,
∂v/∂x = 2y
Therefore,
∂u/∂x = ∂u/∂x + i(2y)

Partial derivative of u with respect to y:
∂u/∂y = ∂/∂y (u + iv) = ∂u/∂y + i∂v/∂y
Since v = 2xy,
∂v/∂y = 2x
Therefore,
∂u/∂y = ∂u/∂y + i(2x)

Step 3: Equating the partial derivatives
Since f(z) is an analytic function, it satisfies the Cauchy-Riemann equations:
∂u/∂x = ∂v/∂y
∂u/∂y = -∂v/∂x

Equating the partial derivatives obtained in Step 2:
∂u/∂x = ∂u/∂y + i(2x) ...(1)
∂u/∂y = -∂u/∂x + i(2y) ...(2)

Step 4: Solving the equations
From equation (1), we have:
∂u/∂x - ∂u/∂y = i(2x) ...(3)

From equation (2), we have:
∂u/∂x + ∂u/∂y = i(2y) ...(4)

Adding equations (3) and (4):
2∂u/∂x = i(2x + 2y)

Simplifying further:
∂u/∂x = i(x + y)

Step 5: Determining the function
Comparing the derived partial derivative with the given function v = 2xy, we can conclude that:
∂u/∂x = i(x + y) = 2xy

Simplifying:
i(x + y) = 2xy

Comparing the coefficients of x and y:
x + y = 2x

Simplifying further:
y = x

Therefore, the correct option that represents the function f(z) = u + iv is option 'A