#CSIRNET #mathematics DEC 2024 | Complex Analysis | Analytic Function

In this video, we are going to discuss the problem of CSI and it December 2024 problem of complex analysis. In the equation we have your complex function f of z equal to e power cos 1 plus i sin z for z is equal to x plus i y we can write f of z as a u x y plus i v x y where u and v are the real value function which of the following is the value of del u by del x at 0 0. So, we have gained 4 option first option is 0 second option is 1 plus 1 upon e cos 1 divided by 2 third option is 1 minus 1 upon e cos 1 divided by 2 and the fourth option is 1. Now, we have gained function f of z equal to e power cos 1 plus i sin z. In the second sentence of the equation, we have to write f of z as a u x y plus i v x y but for the given function it is difficult to write f of z in the form of u x y plus i v x y and moreover we are not interested in the value of u and v but we are interested in the value of del u by del x at 0 0 for that we are going to use the property of the function the given function is exponential function an exponential function is differentiable as well as analytic therefore for any analytic function f of z equal to u x y plus i v x y we can write down f dash z equal to u x x y plus i v x x y where u x is a partial derivative u with respect to x and v x is a partial derivative of v with respect to x now evaluate at 0 so f dash 0 equal to u x 0 0 plus i v x 0 0 and we want the value of u x 0 0 so which is a real part of f dash 0 so for the given function we are going to find the f dash z and we have f dash z equal to e power cos 1 plus i sin z and derivative of cos 1 plus i sin z is cos 1 plus i into cos z because cos 1 plus i is a constant now evaluate at 0 so we have f dash 0 equal to e power cos 1 plus i sin 0 cos 1 plus i sin 0 has value 0 so we have e power 0 cos 1 plus i and cos 0 has value 1 now if you simplify then f dash 0 equal to cos 1 plus i now we try to find the real and imaginary part of cos 1 plus i so for that we are going to use the trigonometric property cos alpha plus beta which is cos alpha cos beta minus sin alpha sin beta here alpha equal to 1 and beta equal to i so we have cos 1 cos i minus sin 1 sin i now to find the value of cos i and sin i just we recall the formula cos z equal to e power i z plus e power minus i z divided by 2 and sin z equal to e power i z now in this formula we replace z by i so in the formula of cos z if we replace z by i then we have e power i into i plus e power minus i into i divided by 2 and in the formula of sin z z replaced by we have e power i into i minus e power minus i into i and divided by 2 i now i into i is i square i square is value minus 1 so we have e power minus 1 plus e power minus i into i minus i square is value 1 so e power 1 similarly e power minus 1 minus e power 1 divided by 2 i now we simplify then f dash 0 equal to cos 1 divided by 2 e plus 1 upon e plus minus 1 equal to i square sin 1 divided by 2 i e power minus 1 minus e power 1 now i square divided by so we have i only and f dash 0 equal to cos 1 divided by 2 e plus 1 upon e plus i into sin 1 divided by 2 1 upon e minus e since f dash 0 is equal to ux 0 plus i vx 0 0 so ux 0 0 is a real part of the f dash 0 which is cos 1 divided by 2 e plus 1 upon e and which is the required answer and which is a option number 2 so option number 2 is the correct answer so this is the simplest way to solve this problem do not require to find the real imaginary part of f of z which is a very difficult task and time consuming part in the csi net exam we don't have that much time to solve the problem so this is a smarter way to solve the problem thanks for watching for more such video please like share and subscribe the next gen mess