 Suppose we've written our complex function as the sum of its real and complex components, and suppose that the derivative exists. So if the limit exists, it must exist regardless of how we approach the point. So suppose we have our complex number and h is real, then we can find our difference quotient. Taking the limit as h goes to zero gives us the derivative of f on the left, and these two become the partial of u with respect to x, and the partial of v with respect to x. Now suppose h is i m, a pure imaginary number. Again, finding our difference quotient will give us, and taking our limit as m goes to zero, we're assuming the derivative exists, so we still have f' of z, and on the right-hand side, we have our partial of u with respect to y, and our partial of v with respect to y. And this gives us two possible expressions for the derivative. Now they're supposed to be the same, so that means we can compare the real and imaginary parts, and that will give us... Consequently, if the derivative exists, we have to have an equality among the partial derivatives of the real and complex parts. These relationships between the derivatives are known as the Cauchy-Riemann equations. Moreover, we can use either of our two expressions as our expression of the derivative, or we could mix and match them. Note that in this form, the Cauchy-Riemann equations are a consequence of differentiability, so if they fail, the function is not differentiable. For example, suppose we want to prove that the real part of z is not differentiable at any point. So let's pull in the Cauchy-Riemann equations. So first, we want to write our function in terms of its real and complex components. So if z is the complex number x plus iy, f of z is x, and this tells us that u of xy is x, and v of xy is 0. Finding the partial derivatives, and even if this is the biggest 0 ever, these two are not equal. And so our function is not differentiable at any value of z. With a little more effort, well, actually a lot more effort, we can prove the converse of our result, provided we add one extra requirement. Suppose our function is split into its real and complex components, and the partial derivatives satisfy the Cauchy-Riemann equations, and these partial derivatives are also continuous at our point. The net is differentiable at the point, and a derivative can be expressed in a number of different ways. So, for example, let's find where the function, the square of the complex conjugate, is differentiable. So let's rewrite our function in terms of its real and imaginary components. If z is x plus iy, then f of z will be, and we can write our function in terms of its real and imaginary components. To satisfy the first of the Cauchy-Riemann equations, the partial of u with respect to x must equal the partial of v with respect to y. So finding these partial derivatives, and so we require, and solving gives us x equal to zero. To satisfy the other Cauchy-Riemann equations, we need to find the other partial derivatives. In order to satisfy the Cauchy-Riemann equations, we require, and solving gives us y equal to zero. Now, remember that differentiability also requires our partial derivatives to be continuous. So, let's verify that. And at x equals zero, y equals zero, these are all continuous functions. And note there's no other place where the Cauchy-Riemann equations will be satisfied, and so f of z is differentiable only at z equal to zero.