All $f\in H(\mathbb{C})$ with $Re(f(z))=u(z)=u(x+iy)=x^2-y^2+2x+1$

Question

In my complex analysis course I'm supposed to compute all entire functions with the given requirement for the Real part. How do I work on this?
I hav no clue how to work this out. Can someone please give me a hint?

Answer 1

$f$ must satisfie the Cauchy-Riemann equations, that's it, $u_x=v_y$ and $v_x=-u_y$, where $v$ is the imaginary part of $f$. So first $$v_y=2x+2\implies v(x, y)=2xy+2y+\phi(x).$$ Also $$u_y=-v_x\implies -2y=-2y-\phi'(y)\implies \phi(y)=+C.$$ So the posibilities to $v$ are $v(x, y)=2xy+2y+C$, and the functions $f$ are $$f(x,y) =(x^2-y^2+2x+1) +i(2xy+2y+C). $$