# Does there exist an analytic function whose real part is $x^2+y^2$?

Does there exist an analytic function whose real part is given by $$u(x, y) = x^2 + y^2$$, where $$z = x + iy$$?

My attempt: I think yes because it will satisfy the Cauchy-Riemann equation; we know that every function that satisfies the Cauchy Riemann equation is analytic.

Is this true?

• You need two functions to test the CR equations. What is your candidate for the imaginary part?
– lulu
Jan 8, 2020 at 20:38
• may be $0$@lulu Jan 8, 2020 at 20:39
• There can't be an analytic function which only takes real values (other than constants).
– lulu
Jan 8, 2020 at 20:41

There is not an analytic function with such property because $$u$$ is not harmonic: $$u_{xx}=2$$ and $$u_{yy}=2$$ so $$u_{xx}+u_{yy}=4\neq 0$$.

There is a result that the real and the imaginary parts of an analytic function must be harmonic (by the way, this follows from Cauchy-Riemann equations).

Any such analytic function $$f=u+iv$$ must satisfy

$$2x=\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$$

and

$$2y=\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$

The first equation gives $$v=2xy+a(y)$$ but the second equation gives $$v=-2xy+b(x)$$. You can then see that there is no such $$v$$.