# Finding the real and imaginary parts of a function

#### shen07

##### Member
If f:C-->C is holomorphic and , find the real and imaginary parts ug and vg of g in terms of the real and imaginary parts uf and vf of f.

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Re: Please can you give me some hint to do this exercise

For clarification you mean by $$\displaystyle u_g=\text{Re}(g)$$ and $$\displaystyle v_g=\text{Im}(g)$$ using that $$\displaystyle g(x,y) = u(x,y)+iv(x,y)$$ , right?

#### shen07

##### Member
Re: Please can you give me some hint to do this exercise

For clarification you mean by $$\displaystyle u_g=\text{Re}(g)$$ and $$\displaystyle v_g=\text{Im}(g)$$ using that $$\displaystyle g(x,y) = u(x,y)+iv(x,y)$$ , right?
yeas right

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Re: Please can you give me some hint to do this exercise

I would suggest starting by

$$\displaystyle u_f = \frac{f(z)+\overline{f(z)}}{2}$$

#### shen07

##### Member
Re: Please can you give me some hint to do this exercise

I would suggest starting by

$$\displaystyle u_f = \frac{f(z)+\overline{f(z)}}{2}$$

One more question what is $$\displaystyle \overline{f(\overline{z})}$$ actually?? i dont quite understand this!

#### topsquark

##### Well-known member
MHB Math Helper
Re: Please can you give me some hint to do this exercise

One more question what is $$\displaystyle \overline{f(\overline{z})}$$ actually?? i dont quite understand this!
Consider a simple example:
$$f(z) = u(z) + i v(z)$$ with z = x + iy.

Then
$$f(z) = u(x + iy) + i v(x + iy)$$

$$f( \overline{z} ) = u(x - iy) + i v(x - iy)$$

$$\overline{f( \overline{z} ) } = u(x - iy) - i v(x - iy)$$

Is this what you are looking for? Or something more conceptual?

-Dan

Last edited:

#### Fernando Revilla

##### Well-known member
MHB Math Helper
Another example:
$$f(z)=z^2+i \Rightarrow \overline{f\left({\bar{z}}\right)}=\overline{(\bar{z})^2+i}=\overline{ \overline{z^2}+i}=\overline{\overline{z^2}}+\bar{i}=z^2-i$$