 Hello everyone. In the last two videos we have discussed about the definitions and some examples. In this video also we will consider few more examples. The learning outcome of this lesson is at the end of this session students will be able to verify CR equations and analyticity of a given complex valued function. Consider the example. Verify f of z equal to e raised to z is analytic or not. Solution. Write the given function f of z as u plus i v and it is given as e raised to z and we know that z equal to x plus i y. Therefore replacing this z by the expression x plus i y we get it as e raised to x plus i y. That is u plus i v equal to again we are knowing the property of the exponential that e raised to a plus b can be written as e raised to a into e raised to b. After using this identity here we get e raised to x plus i y as e raised to x into e raised to i y. Again we are knowing that e raised to i theta can be written in the complex number form cos of theta plus i sin theta. Applying this result for this e raised to i y we get e raised to x into cos y plus i sin y. Let us multiply by e raised to x to this bracket we get it as e raised to x into cos y plus i into e raised to x into sin y. Now let us equate real and imaginary parts from the both sides we get u equal to e raised to x into cos y and v equal to e raised to x into sin y. Now this is the u and v. Now in order to check whether the given function is analytic or not we will discuss the sufficient condition for the analyticity. The first condition is Cauchy-Riemann equations. Now in order to have the Cauchy-Riemann equations let us differentiate this u partially with respect to x and y. Now we will denote in short the partial derivative of u with respect to x as the symbol u x. Now here while differentiating partially with respect to x we have to treat y constant for this cos y is constant and the derivative of e raised to x is e raised to x itself. Again differentiating u partially with respect to y we get as we are differentiating with respect to y x we have to treat constant. So e raised to x as it is and the derivative of cos y is minus sin y and the final answer is minus e raised to x into sin y. Similarly the partial derivative of v with respect to x that is vx is equal to e raised to x into sin y and the partial derivative of v with respect to y is obtained as e raised to x into cos y. Now here we can see that ux and vy are having the same expression e raised to x into cos y. Therefore we can write ux equal to vy and it is equal to e raised to x into cos y. Again we can see that uy and vx both are equal there is difference between the sin only. So we can write here uy is nothing but minus of vx and it is nothing but minus e raised to x into sin y. Now from these two equalities we can say that the given function satisfies CR equations. Now let us consider the second condition that is the continuity of these four partial derivatives. Now these are the partial derivatives. Now we can see that exponential cos and sin are continuous functions. And again we know that the product of continuous functions is again a continuous function. By using this property here we can see that all these four derivatives are obtained in terms of the product of the continuous functions. So obviously all the partial derivatives are continuous conditions of sufficient one are satisfied. Therefore we can say that the given function f of z equal to e raised to z is analytic everywhere. That is we can say that it is the entire function. Now pause this video and verify the analyticity of the function f of z equal to mod z square. I hope all of you have written an answer. Now the problem is to check the function mod z square is analytic or not. Let us denote this function by f of z which has the expression u plus iv. And we know that z is a complex number and it has the form x plus iv. And the meaning of this symbol is the modulus of z. And it is defined as under root of x square plus y square. That is under root of square of the real part plus square of the imaginary part. Now we can insert a bracket and square and the definition of mod z is root of x square plus y square. And square of a square root is this expression itself x square plus y square. Now here we can see that this f of z here we have obtained as a real valued function. And we know that every real valued function can be written as a complex valued function by writing plus i into 0 as the imaginary part. Now it will look like a complex valued function. Now let us compare the real and imaginary parts from the both sides. We get the real part u equal to x square plus y square and the imaginary part v equal to 0. In order to check the analyticity of given function, let us use the sufficient condition. The first condition is of CR equations. Now let us differentiate this u partially with respect to x treating y constant. We get it as ux equal to the derivative of x square is 2x and as y constant its derivative is 0. Again differentiating this u partially with respect to y treating x constant. Now the derivative of x is 0 and the derivative of y square is 2y. As v is 0 so obviously the partial derivative of this v with respect to x as well as y both are 0. Now here we can see that this ux and vy, ux is 2x and vy is 0 and uy is 2y and vx is 0. They are not equal. As the CR equations are not satisfied but for the analyticity of any function CR equation is the necessary one. Therefore as the necessary condition fails here we can say that f of z is not analytic at any point in a given complex plane. And those functions which are not analytic at a single point in a complex plane such functions are called as nowhere analytic. Therefore the given function f of z is nowhere analytic. Let us consider one more example. Determine whether 1 upon z is analytic or not. Let us denote the given function by f of z which has the form u plus iv and it is equal to 1 upon z. Now it will implies that u plus iv equal to we know that z is nothing but x plus iv. Now this is not looks like in a complex function form. So we will convert it to a complex function form by multiplying and dividing by the conjugate of this denominator complex form. The conjugate of this denominator is x minus iv. Let us multiply by x minus iv and divide by this x minus iv. We get it as x minus iv upon. Again we know that the product of a complex number and its conjugate is nothing but the sum of the square of its real part and the imaginary part. That is we get it as x square plus y square. Let us separate it from the numerator. We get u plus iv equal to x upon x square plus y square minus i into y upon x square plus y square. Let us equate the real and imaginary part. We get u equal to x upon x square plus y square. We equal to minus y upon x square plus y square. Again to check the analyticity we are first of all checking the CR equations. To check the CR equations let us differentiate this u partially with respect to x. We get dou u by dou x and it is equal to. Now here we can see that u is we have obtained in terms of a rational function. And this rational function can be differentiated by using quotient rule as x is involved in the numerator and denominator. So, writing denominator as it is and differentiating numerator partially with respect to x. We get it as 1 minus numerator x as it is and the derivative of denominator is 2x divided by square of the denominator. That is x square plus y square whole square. And after simplifying we get it as y square minus x square upon x square plus y square bracket square. Again differentiating the same u partially with respect to y treating x constant that is dou by dou y of. Here u is x upon x square plus y square now y is involved in the denominator. So, we can directly differentiate it and as x is constant we can write it as x outside the operator x into dou by dou y of 1 upon x square plus y square. And its derivative is given by x as it is and the derivative of 1 upon x square plus y square is minus 1 upon x square plus y square bracket square. Into partial derivative of this x square plus y square with respect to y is 2y and finally we get it as minus 2xy upon x square plus y square bracket square. Similarly differentiating v partially with respect to x we get it as 2xy upon x square plus y square bracket square and dou v by dou y as y square minus x square upon x square plus y square bracket square. Now here we can see that from 1 and 4 dou u by dou x is exactly equal to dou v by dou y and from 2 and 3 dou u by dou y is equal to minus of dou v by dou x. That is it implies that the Cauchy-Riemann equations are satisfied. And again we can see that all these four derivatives are obtained in terms of the polynomials in the numerator and denominator and we know that the polynomials are continuous functions. But as the denominator contains x square plus y square only and each and every derivative at point 0 comma 0 these functions are not defined therefore they are not continuous at that point. It says that except at point 0 0 all the four partial derivatives are continuous everywhere. Therefore the condition 1 and 2 is satisfied except at point z equal to 0. Therefore we can say that the given function f of z equal to 1 upon z is analytic everywhere in the complex plane except at point z equal to 0.