Real and Imaginary Parts of a Complex Product

**bkarpuz** · October 5th 2010, 06:36 AM

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?

**emakarov** · October 5th 2010, 11:17 AM

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 $\varphi(k)$ .

**bkarpuz** · October 5th 2010, 12:13 PM

Thanks for the information but this seems to be useless for me. :s

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

Thread: Real and Imaginary Parts of a Complex Product

LinkBack

Thread Tools

Display

Real and Imaginary Parts of a Complex Product

Similar Math Help Forum Discussions

kindly explain this step of real and imaginary parts

[SOLVED] Real and Imaginary parts

real and imaginary parts of tan(z)

I need help with complex numbers and real and imaginary.

Real / Imaginary Parts

Search Tags