For a function f(z)=u(x,y)+iv(x,y) to be analytic then the C-R equation to be satisfied are

The Cauchy-Riemann (C-R) equations are a set of two partial differential equations that are necessary and sufficient for a function to be analytic. For a function f(z) = u(x, y) + iv(x, y), where u and v are real-valued functions of two real variables, x and y, the C-R equations are:

1. ∂u/∂x = ∂v/∂y
2. ∂u/∂y = -∂v/∂x

These equations state that the partial derivatives of u and v must satisfy these relationships for the function f(z) to be analytic.

Here's a step-by-step breakdown:

Step 1: Identify the real and imaginary parts of the function. In this case, u(x, y) is the real part and v(x, y) is the imaginary part.

Step 2: Compute the partial derivatives of u and v with respect to x and y.

Step 3: Check if the first C-R equation is satisfied, i.e., if the partial derivative of u with respect to x equals the partial derivative of v with respect to y.

Step 4: Check if the second C-R equation is satisfied, i.e., if the partial derivative of u with respect to y equals the negative of the partial derivative of v with respect to x.

If both equations are satisfied, then the function f(z) is analytic.