1. ## anlytic function

How can I find all analytic functions f=u+iv with u(x,y)=(x^2)+(y^2)

Thanks for the help. I appreciate it.

2. ## Re: anlytic function

a) do you know what it means for a function to be analytic?
b) do you know what the Cauchy–Riemann equations are?

if not read up on those two and give it a shot.

3. ## Re: anlytic function

i tackled with the problem which is u(x,y)=(x^2)-(y^2)

but i need help for that one u(x,y)=(x^2)+(y^2)

4. ## Re: anlytic function

to the best of my knowledge
if u(x,y)=(x^2)-(y^2) then

f=(x^2)-(y^2)+ i*2xy

5. ## Re: anlytic function

Originally Posted by mami
i taclked the problem which is u(x,y)=(x^2)-(y^2)

but i need help for that one u(x,y)=(x^2)+(y^2)

ok, so you know that du/dx = dv/dy and du/dy = -dv/dx

you have u so compute it's partials

du/dx = 2x
du/dy = 2y

so dv/dy = 2x and thus v = 2xy + f(y)

dv/dx = -du/dy = -2y so v = -2xy + g(x)

2xy + f(y) = -2xy + g(x)

4xy = g(x) - f(y), and you can see that this has no solution

6. ## Re: anlytic function

thanks again. but I have a problem again. could you please check it. is it relevant our question? my mind confused...

7. ## Re: anlytic function

at (0,0) 4xy = 0 so the C-R equations happen to have a solution there so yes (x^2+y^2) + i*0 happens to satisfy the C-R equations at 0.

so yeah, you can say (x^2+y^2) + i 0 is analytic at 0 but nowhere else on the complex plane.

8. ## Re: anlytic function

would you mind, may i send e-mail to you if i couldn't tackle any problem?

9. ## Re: anlytic function

Originally Posted by romsek
at (0,0) 4xy = 0 so the C-R equations happen to have a solution there so yes (x^2+y^2) + i*0 happens to satisfy the C-R equations at 0.

so yeah, you can say (x^2+y^2) + i 0 is analytic at 0 but nowhere else on the complex plane.
I should be a bit more careful. You need more than satisfying the C-R equations to declare a function is analytic at a point. The first partials must exist and be continuous at that point. The first partials of (x^2+y^2) + i*0 do exist and are continuous everywhere and thus are at (0,0).

10. ## Re: anlytic function

posting them will be faster as I'm not always around.