# Real and Imaginary Parts of a Complex Product

• Oct 5th 2010, 07:36 AM
bkarpuz
Real and Imaginary Parts of a Complex Product
Dear MHF users,

let $\varphi:\mathbb{N}\to\mathbb{R}$ be a function and $n\in\mathbb{N}$.
For $z\in\mathbb{C}$ define a function $\Phi:\mathbb{C}\to\mathbb{C}$ by
$\Phi(z):=\prod_{k=1}^{n}\big(1+\varphi(k)z\big)$.
How can I decompose this type of functions into its real and imaginary parts?
• Oct 5th 2010, 12:17 PM
emakarov
If you could represent $1+\varphi(k)z$ as $r_ke^{i\alpha_k}=r_k(\cos\alpha_k+i\sin\alpha_k)$, then the product is $(\prod_{k=1}^nr_k)(\cos(\sum_{k=1}^n\alpha_k)+i\si n(\sum_{k=1}^n\alpha_k))$.

Now, if $z=re^{i\alpha}$, then $\varphi(k)z=\varphi(k)re^{i\alpha}$. Also, using some trigonometry, one can express $1+\varphi(k)z$ as $r_ke^{i\alpha_k}$ where $r_k$ and $\alpha_k$ are expressions containing $r$, $\alpha$ and $\varphi(k)$.
• Oct 5th 2010, 01:13 PM
bkarpuz
Thanks for the information but this seems to be useless for me. :s

But but but let me think it may give a hand…