Finding the real and imaginary parts of a function

[shen07]

If f:C-->C is holomorphic and View attachment 1263 , find the real and imaginary parts u_g and v_g of g in terms of the real and imaginary parts u_f and v_f of f.

 

[alyafey22]

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]

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

 

[alyafey22]

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]

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 don't quite understand this!

 

[topsquark]

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

One more question what is \(\displaystyle \overline{f(\overline{z})}\) actually?? i don't 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

 

[Fernando Revilla]

Another example:
$$f(z)=z^2+i \Rightarrow \overline{f\left({\bar{z}}\right)}=\overline{(\bar{z})^2+i}=\overline{ \overline{z^2}+i}=\overline{\overline{z^2}}+\bar{i}=z^2-i$$