8. For the following functions show that f'(z) does not...

8. For the following functions show that f'(z) does not exist as CR equations are not satisfied. i. f(z) = ? ii. f(z) = z - ? iii. f(z) = 2x + ixy^2 iv. f(z) = e^x e^{-iy} 9. Show that f'(z) and its derivative f''(z) exist every-where and find f''(z) when (a) f(z) = iz + 2 (b) f(z) = e^{-x}e^{-iy} (c) f(z) = z^3 (d) f(z) = cos x cosh y - i sin x sinh y. 10. Determine where f'(z) exists and find its value when; (a) f(z) = 1/z (b) f(z) = x^2 + iy^2 (c) f(z) = z Im z. 11. Show that the function f(z) = x + 4iy is nowhere differentiable. 12. Show that the function f(z) = z^2 + z is analytic. 13. Show that the function f(z) = 2x^2 + y + i(y^2 - x) is not analytic.

Transcript

00:02 Hello, let f be a complex valued function.

00:06 It's a function from c to c and it can represent the function as u plus iv and the cauchy -riemann equations are ux equal to vy and uy is equal to minus vx.

00:21 In question 8, we have to show that none of these four functions satisfy the cauchy -riemann equation at any point.

00:27 So all this each of these four functions will not be differentiable at every point of c.

00:35 So fg the first function is z bar which is x plus i times minus y.

00:43 So this is u, this is u and this is v.

00:47 Now ux is equal to 1 and vy is equal to minus 1.

00:55 Since ux is not equal to vy, this does not it at no point ux can be equal to vy.

01:02 This does not satisfy cauchy -riemann equation at any point.

01:09 The second function is z minus z bar which we can which when we simplify it is 0 plus i times 2y.

01:18 This is v and 0 is the u.

01:22 Then ux is 0, vy is 2 and they can never be equal at any point.

01:27 So this function does not satisfy c -r equations at any point.

01:32 So it is not differentiable at any point.

01:36 Now the third function is fz is equal to 2x plus i xy square.

01:41 This is u and this is v.

01:44 Ux is 2, vy is 2xy and uy is 0 and vx is y square.

01:56 Now if this if uy equal to minus vx, the second cauchy -riemann equation, then we must have that y square equal to 0 or y equal to 0.

02:10 But then if y equal to 0 then vy is 0 but ux is always 2.

02:16 So then this equation is not satisfied.

02:19 So the two equations, these two equations are not satisfied at any point.

02:27 For the fourth function fz is equal to e to the power x times e to the power minus iy.

02:35 E to the power minus iy is cos minus y plus i sin minus y.

02:41 Cos minus y is cos y.

02:43 Sin minus y is minus sin y.

02:45 So e to the power minus iy is cos y minus i sin y.

02:52 So we can write this as e to the power x, this whole thing as e to the power x cos y plus i times minus e to the power x sin y.

03:01 This is u and this is v.

03:04 Ux is e to the power x cos y.

03:06 Vy is minus e to the power x cos y.

03:12 And uy is minus e to the power x sin y.

03:16 And vx is minus e to the power x sin y.

03:21 Now if these two things have to be equal it will mean that we cancel minus, sorry, if this has to be minus of this then we can cancel this minus and e to the power x.

03:37 E to the power x is always non -zero so we can cancel minus e to the power x...

Question

Please give Ace some feedback