Find an analytic function from its real part

Hi there. I have two doubts on this exercise. The exercise gives me the real part u(x,y), and I have to find from it the imaginary part, such that the function I get accomplishes the given condition, and its an analytic function.

The first doubt I got I think its quiet silly. I don't know how I should evaluate f(i). I hoped to have there two coordinates, for x and y. But I have just i. I don't know if I should took that as a complex number expressed in its polar form, does it mean that x=0 and y=i?

On the other hand I've tried to find v(x,y), the real part it asks me for.

So I used the Cauchy-Riemann conditions.

So there is something wrong in there, the function I get for Phi depends on y too. Whats wrong with this? I couldn't find the analytic function under this procedure. Did I do something wrong or is it that it doesn't exist an analytic function for the given u(x,y)?

Bye and thanks.

PD: Ok, I realized about the first part, it was really silly, it just means f(z)=f(i), z=i.

Re: Find an analytic function from its real part

Quote:

Originally Posted by

**Ulysses** Hi there. I have two doubts on this exercise. The exercise gives me the real part u(x,y), and I have to find from it the imaginary part, such that the function I get accomplishes the given condition, and its an analytic function.

The first doubt I got I think its quiet silly. I don't know how I should evaluate f(i). I hoped to have there two coordinates, for x and y. But I have just i. I don't know if I should took that as a complex number expressed in its polar form, does it mean that x=0 and y=i?

On the other hand I've tried to find v(x,y), the real part it asks me for.

So I used the Cauchy-Riemann conditions.

So there is something wrong in there, the function I get for Phi depends on y too. Whats wrong with this? I couldn't find the analytic function under this procedure. Did I do something wrong or is it that it doesn't exist an analytic function for the given u(x,y)?

Bye and thanks.

PD: Ok, I realized about the first part, it was really silly, it just means f(z)=f(i), z=i.

Your porcedure is correct!... from the condition...

(1)

... You derive immediately that is...

(2)

... so that is...

(3)

Taking into account that You obtain in (3) ...

Kind regards