 Welcome back MechanicalEI! Did you know that all elementary functions like polynomials, exponential functions, trigonometric functions, absolute values, etc. are analytic functions? This makes us wonder, what are analytic functions? Before we jump in, check the previous part of this series to learn about what complex variables are. Now, a function f of z is said to be analytic in a region r of complex plane if f of z has a derivative at each point of r and if f of z is single-valued. Also, a function f of z is said to be analytic at a point r if z is an interior point of some region where f of z is analytic. That is, the concept of analytic function at a point implies that the function is analytic in some circle with its center at this point. Let f of x, y equals to u of x, y plus i into v of x, y be a complex function since x equals z plus z bar upon 2 and y equals to z minus z bar upon 2i. Substuting for x and y gives f of z comma z bar equals u of x comma y plus i into v of x comma y. A necessary condition for f of z comma z bar to be analytic is dou f upon dou z equals 0. Mark it as 1. Therefore, a necessary condition for f equals to u plus i v to be analytic is that f depends only on z. In terms of the real and imaginary parts u comma v of f, condition 1 is equivalent to dou u upon dou x equals dou v upon dou y. Mark this as 2. And dou u upon dou y equals minus dou v upon dou x. Mark this as 3. Equations 2 and 3 are known as Cauchy-Riemann equations. They are a necessary condition for f equals u plus i v to be analytic. The necessary and sufficient conditions, however, are two in numbers. First, the four partial derivatives of real and imaginary parts should satisfy the Cauchy-Riemann equations. And second, the four partial derivatives of its real and imaginary parts should be continuous. Hence, we first saw what analytic functions are and then went on to see the necessary and sufficient conditions for it. So, like, subscribe and comment with your feedback to help us make better videos. Thanks for watching. Also, thanks a lot for those constructive comments. You helped the channel grow. So, here are the top mechanical EIs of our last videos. In the next episode of Mechanical EI, find out what Milner-Thompson methods are.