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On $U=\mathbb{C}-\{0\}$, find a holomorphic function $f=u+iv$.

$$

u(x,y) = \frac{y}{x^2+y^2}

$$

u is harmonic on U

Let g be a primitive for f on U.

write $g=\varphi +i\psi$.

Then $\varphi_x = u$.

$$

\varphi_{xy} = \psi_{xy} = \frac{x^2-y^2}{(x^2+y^2)^2}

$$

So I can integrate the above with respect to x and find a function with the constant of integration being some h(y).

Then I would have v and I would have found my function f correct?

So I found $f$ to be

$$

f(z) = u + iv = \frac{y}{x^2 + y^2} + i\left[\frac{x}{x^2 + y^2} + h(y)\right].$$

Correct?

$$

u(x,y) = \frac{y}{x^2+y^2}

$$

u is harmonic on U

Let g be a primitive for f on U.

write $g=\varphi +i\psi$.

Then $\varphi_x = u$.

$$

\varphi_{xy} = \psi_{xy} = \frac{x^2-y^2}{(x^2+y^2)^2}

$$

So I can integrate the above with respect to x and find a function with the constant of integration being some h(y).

Then I would have v and I would have found my function f correct?

So I found $f$ to be

$$

f(z) = u + iv = \frac{y}{x^2 + y^2} + i\left[\frac{x}{x^2 + y^2} + h(y)\right].$$

Correct?

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