I need to find $\operatorname{Im} f$

Question

Let $f$ be analytic such that $\operatorname{Re} ( f'(z))=2y$ and $f(1+i)=2$. I need to find $\operatorname{Im} f$.

I took $f(z)=u(x,y)+i v(x,y)$ but I don't know $\frac{df(z)}{dz}=$ ? in terms of $u(x,y)$ and $v(x,y)$.

Could any one help me in that step?

score 5 · Answer 1 · answered Sep 03 '13 at 14:27

5

Hint: First find $f'(z)$ by the usual method of computing harmonic conjugates. (Just use the Cauchy-Riemann equations.) Then integrate to find $f(z)$ up to a constant of integration. Use your initial condition $f(1+i)=2$ to determine $f(z)$ uniquely.

answered Sep 03 '13 at 14:27

Potato

40,171

$f'(z)=-iu_y+v_y=u_x+iv_x$ what next? – Myshkin Sep 03 '13 at 14:47
@miosaki Well, you know $u$. So you know $u_x$ and $u_y$. This gives you $v_y$ and $v_x$. Integrating $v_x$ and $v_y$ gives you enough information to find $v$. – Potato Sep 03 '13 at 14:49
I know only Real part of $f'(z)$ which is $2y$ I dont know $u(x,y)$ – Myshkin Sep 03 '13 at 14:52
$u_x(x,y)= 2y$ ???? – Myshkin Sep 03 '13 at 14:52
$v_y(x,y)=2y$??????? – Myshkin Sep 03 '13 at 14:53
@miosaki Sorry, I was using the following notation: $f'=u+iv$. You first solve for $f'$, forgetting about the derivative, then integrate to find $f$. – Potato Sep 03 '13 at 14:53
I am not able o follow please do it – Myshkin Sep 03 '13 at 15:09
@miosaki You know $u=2y$. So $u_x=0=v_y$ and $u_y=2=-v_x$. Solving this differential equation allows you to find $v$. Then integrating $u+iv$ gives you $f(z)$. – Potato Sep 03 '13 at 15:18

Ömer · Accepted Answer · 2013-09-03T15:42:36.273

1

$$f'(z)=u_x+iv_x=v_y-iu_y$$ If $u_x=2y$ then $u=2xy+h(y)$ and if $v_y=2y$ then $v=y^2+g(x)$ .So $$f(z)=2xy+h(y)+(y^2+g(x))i$$ we know also $f(1+i)=2.1.1+h(1)+(1^2+g(1))i=2$ and $v_x=-u_y$. Hence $h(y)=0$ and $g(x)=-x^2$. Then $f(z)=2xy+(y^2-x^2)i$. Therefore $Im(f)=y^2-x^2$.

edited Sep 03 '13 at 15:42

answered Sep 03 '13 at 15:29

Ömer

2,038

I need to find $\operatorname{Im} f$

2 Answers2