 In complex analysis, a function could be differentiable at a point or at every point in an open region. And so we define the following. Suppose the derivative of a function exists at all points in some open region, r, then the function is analytic in r. Now, a useful idea to keep in mind is the more words we have for the same thing, the more important it probably is. So, while we'll use the term analytic, we also use the terms holomorphic or regular. And if the open region includes all of the complex plane, we say the function is entire. So why do we care? Don't jump your cue! The terms are interchangeable and really tell you more about who someone learned complex analysis from than anything else. So suppose a function is analytic in some region, then it has a first derivative for all complex numbers in that region, and so you've got to ask yourself what would be required for it to have a second derivative for at least some of the complex numbers in that region. So let's suppose our function is split into its real and complex parts, and suppose it's analytic at some point. By the Cauchy-Riemann equations, we know the relationship between the derivatives and an expression for the derivative itself. If we want f' to also be differentiable, we need the partial derivatives to be continuous, and for u and v to satisfy the Cauchy-Riemann equations. Now remember capital U and capital V are already partial derivatives, so for there to be even a chance for this to work, our second partial derivatives have to exist. But we have no guarantee that's true. Well that's okay. Remember you can have anything you want as long as you pay for it, and in this case we'll just make that assumption. So we want our partial derivatives to satisfy the Cauchy-Riemann equation. So we want to start with a partial of u with respect to x and show that it's equal to the partial of v with respect to y. So strategically because we have a partial with respect to x here and a partial with respect to y here, we want to have something with a partial with respect to the other variable someplace. And so by the Cauchy-Riemann equations we know that u is the partial of v with respect to y. And we can differentiate it, and well we're kind of stuck. So let's work the other end. We also know that v is the partial of v with respect to x. And we can work our way backwards a step. And now we have these mixed partials. But since we assume v was continuous in the region, the mixed partials are also equal. So the first of the Cauchy-Riemann equations is satisfied. By essentially the same argument, we can show that the derivative of u with respect to y is the negative of the derivative of v with respect to x. And this leads to an important result. Suppose f of z is analytic at z naught, then f prime is also analytic at z naught. Now if you've read the proof carefully you'll know there is an assumption there that our partial derivatives exist and are continuous. So we should say that although as it turns out the property of being analytic is such a strong property that it actually makes our partial derivatives exist and be continuous. So we don't actually need to claim this as part of our theorem. It's a result of, well, another theorem. And now we can lather Rind's repeat. Since our derivative is analytic at z naught, then the second derivative is also analytic at z naught. And so derivatives of all orders exist at z naught. And this means that we can form the Taylor series for our function around z equals z naught. Now with some effort we can show this series actually converges to the function and this leads to an important result. If a function is analytic at a point it converges to its Taylor series in an open neighborhood around that point. Now we know that the converse is also true and relatively straightforward to prove and that means this can be used as an alternate definition of what it means for a function to be analytic. And in fact it's more useful to think about an analytic function as one that can be defined in terms of its Taylor series.