Question

Confuse!! Please, help
Problem: Where is the function differentiable? \(x^2+y^2 +2ixy\)
Where is it analytic?

Answer 1

\(U_x = 2x = V_y \\U_y = 2y = -V_x = -2y \iff y =0\)
Hence the function is differentiable on x axis.
My question: is it not that it is analytic on x axis also? why the answer is : "No where" ?

Answer 2

hmm you sure this is right?

Answer 3

What is your suspicion?

Answer 4

checking the partials although it can't be that hard, i think it must be of by a minus sign somewhere

Answer 5

\(U(x,y) = x^2 +y^2 \\V(x,y) = 2xy\)

\(U_x = 2x\\U_y= 2y\\V_x = 2y\\V_y = 2x\)
I am sure about that.

Answer 6

then you are off by a minus sign for cauchy riemman right?

Answer 7

\(U_x = V_y=2x\\U_y = -V_x\iff 2y = -2y \iff y =0\)
I am sure about that also.

Answer 8

Yes,

Answer 9

for cauchy riemann \(U_y=-V_x\)

Answer 10

Yes

Answer 11

but is isn't

Answer 12

Hence, the conclusion is the function is differentiable when y =0

Answer 13

That is it is differentiable on Real axis, right?

Answer 14

yeah on the real axis it is \(f(x)=x^2\)

Answer 15

But for part b) Where is it analytic,
From above it is differentiable on Real, it should be analytic on Real also, but the answer is "No where"

Answer 16

It is not f (z)

Answer 17

differentiable means the derivative exists from any direction

Answer 18

f(z) = x^2 +y^2 +2ixy,

Answer 19

and the whole thing differentiable at the point on Real axis only. We can't take off y from f, right?

Answer 20

no
just like on the real line differentiable means the limit exists from the left and the right, in complex plane the limit must exist from any direction

Answer 21

btw if they didn't mention it, as i remember correctly "analytic" means differentiable, but also means you can write the function as a function of \(z\)