 Hello everyone, myself As Falmari, in this video I will explain Cauchy-Riemann equations. The learning outcome of this lesson is at the end of this lesson students will be able to verify Cauchy-Riemann equations and analyticity of a given complex valued function. Let us start this lesson with the definition of a differentiation. A complex valued function f of z equal to u of x comma y plus i v of x comma y in short it can be written as u plus i v is said to be differentiable at a point z equal to z naught if limit of the quantity f of z minus f of z naught divided by z minus z naught as z tends to z naught exist and this limit is independent of the path along which z tends to z naught and if the derivative is exist then it is denoted by f dash of z naught. The meaning of this statement limit is independent of the path along which z tends to z naught is let us consider a complex plane let us assume that this is a point z naught now here we can see that there are so many ways along which we can approach to this z naught the certain paths for approach of z naught are the meaning of that condition is that if we take the limit along any one of the path the limit must be unique if it is the case then we can say that the derivative of the function f at this point z naught exist if the limit is varying that is the limit is not unique along the various paths at that time we can say that the given function is not differentiable at this point z naught one very important property of the differentiability differentiability of f of z implies continuity of f of z but the converse that is continuity of f of z may not imply differentiability of f of z next definition of analytic function first analytic at a point a function f of z is said to be analytic at a point z equal to z naught if it satisfies two conditions the first condition is f of z is differentiable at this point z equal to z naught and the second condition is that there exists at least one neighborhood or one can say some neighborhood say and around the point z naught in which this f of z is differentiable let us understand this definition geometrically this is a complex plane and let us consider this point as a z naught to say that the function f of z is analytic at this point the first condition the function must be differentiable at this point and the second condition is that around this point z naught we can find one neighborhood at least in complex analysis the neighborhood is nothing but a circular disc like this with some radius delta we can find at least one such a circular disc in which the given function f of z is differentiable if this is the case we can say that the given function f of z is analytic at a point z naught if it is not possible to find such a kind of at least one neighborhood in which the function is not differentiable we can say that the function is not analytic at this point z naught next analytic a region d a function f of z is said to be analytic in a domain d if it is analytic at each point of a region d that is geometrically so this is a complex plane and suppose that this is the given region d in which the function is defined a function f of z is said to be analytic in this region if it is analytic at each and every point of this region an analytic function is also known as holomorphic or regular or monogenic function one more definition entire function a function f of z is said to be entire if f of z is analytic at each point of the complex plane one very important note an entire function is always analytic differentiable and continuous functions but the converse is not true and second one is analytic function is always differentiable and continuous function but the converse is not true now just we have discussed about the definition of differentiability and analyticity the analytic of a function is totally depends upon the derivative and as we have seen it is not an easy task to find out the derivative of a function at a given point so obviously to verify whether the given function is analytic at a point in a region is also a difficult task now here we are considering the necessary and sufficient condition for the analyticity by which our work to show that or to verify whether the function is analytic or not is somewhat easy now first of all we will deal with the necessary condition for f of z to be analytic the theorem is the necessary condition for a function f of z is equal to u plus iv to be analytic at all at all the points in a region dr are it must satisfies these two equations that equations are dou u by dou x equal to dou v by dou y and second is dou u by dou y equal to minus of dou v by dou y and obviously provided that all these four partial derivatives exist that is for the analyticity of any function these two equations are necessary to satisfy and these two equations are called as Cauchy-Riemann equations now this is what this is what the necessary condition but not a sufficient condition this condition most of the time is used to show that the given function is not an analytic function because it is a necessary condition not a sufficient one pause the video and answer the question is the function f of z equal to real part of z is analytic I hope all of you have written the answer now the question is to check f of z equal to real part of z is analytic or not now given function is f of z it has the general form u plus iv and in this case it is given as real part of z since we know that z has the general form x plus iv where x is the real part and y is the imaginary part therefore real part of z is x now let us write this x number in terms of the complex number form x plus i into 0 now by the definition of equality of two complex number equating its real and imaginary parts we get real part u equal to x and v equal to 0 now to check whether the given function is analytic or not let us verify the necessary condition of the analyticity which is the cr equations in order to check the cr equations are satisfied or not let us differentiate u partially with respect to x treating y constant therefore we get u x equal to the derivative of x is 1 as u is not containing y therefore the partial derivative of u with respect to y treating x constant is 0 as v is 0 its partial derivative with respect to x as well as with respect to y both are 0 now here we can see that u x is 1 and v y is 0 therefore u x is not equal to v y and which is the one of the condition of cr equations and which is here not equal therefore as the necessary condition for the analyticity is not satisfied we can say that the given function f of z equal to real part of z is not an analytic function now what is the sufficient condition for the function f of z to be analytic the sufficient condition for a function f of z equal to u plus iv to be analytic at all points in a given region dr the first condition is u and v must satisfy the Cauchy Riemann equations that are written in terms of the equalities and obviously provided that all the four partial derivatives exist and second condition is all the four partial derivatives must be continuous functions of x and y in a given region d now these are the two sufficient conditions for the analyticity of any given function f of z that is those functions which satisfies these two conditions that functions are analytic some important notes if a function is analytic in a domain d then u comma v satisfies Cauchy Riemann equations in a region d Cauchy Riemann equations are necessary conditions but they are not sufficient for the analyticity of any function Cauchy Riemann conditions are sufficient for analytic if the partial derivatives are continuous and these are the references