Evaluating complex multiplication?

[Loren Booda]

How might one comprehend the product of complex numbers

N
_[pi](a_n+ib_n)=C
n=1

such as by representation with vectors in the complex plane, or algebraic simplification? Specifically, I would like to know the value Re(C).

 

[sridhar_n]

Hi

Any complex number a+ib can be represented as R(& theta). Where R is the magnitude and & theta is the phase. So convert the complex number into R(& theta) form and multiply. The R parts multiply while the angle parts add up.

e.g. R(& theta)* P(& Theta) = R*P(& theta + & Theta)

Got it?


Sridhar

 

[Loren Booda]

A bit rough, sridhar, but helpful in jogging my memory.

Can you or another be more mathematical in regard to the transform involved?

Is it tan^-1(b/a)=_[the] and r=(a²+b²)^1/2?

 

[Hurkyl]

Right...ish

if (a + bi) = r exp(iθ), then it is true that

r = (a^2 + b^2)^(1/2)
and
tan θ = b/a

But you have to make sure that θ is in the correct quadrant. (iow you might have to add π).

 

[Loren Booda]

Hurkyl,

Is there a simplifying (exact) identity for the arithmetic series

N
[sum]tan^-1(b_n/a_n)
n=1

and for the geometric series

N
_[pi](a_n²+b_n²)^1/2
n=1

or, more importantly, for my original statement concerning Re(C)?