How do I separate the real and imaginary parts of an infinite product?

[benorin]

TL;DR

Just polling to see if anybody knows

Suppose you have a complex-valued function of a complex variable (namely, $z=x+iy, \, \, x,y\in \mathbb{R}$) defined as the assumed convergent infinite product

$$F(z)=\prod_{k=1}^{\infty}f_{k}(z)$$

Further suppose $F(x+iy)=u(x,y)+i v(x,y)$, where u and v are real-valued functions.
How to write “closed form” expressions for u and v in terms of the $f_k$?

 

[andrewkirk]

It depends what you mean by closed form. An infinite product would generally not be considered a closed form.
Perhaps what you are after are expressions for $U_k(z)$ and $V_k(z)$, which are the real and imaginary parts of the partial product $F_k(z)\triangleq \prod_{j=1}^k f_k(z)$. If the infinite product in the OP is convergent then we would expect $U_k$ and $V_k$ to be convergent, to functions $U,V:\mathbb C\to \mathbb R$, such that $F(z)=U(z)+iV(z)$.
Closed forms for the partial products are:
$$U_k(z) = \sum_{s\in \{0,1\}^k} r_u(h(s))\prod_{j=1}^k
u_j(z)^{1-s_j} v_j(z)^{s_j}$$
$$V_k(z) = \sum_{s\in \{0,1\}^k} r_v(h(s))\prod_{j=1}^k
u_j(z)^{1-s_j} v_j(z)^{s_j}$$
where $h(s) = \left(\sum_{j=1}^{\mathrm{length}(s)} s_j\right) \mod 4$
$r_u(m)$ is 1 if $m=0$, -1 if $m=2$ and 0 otherwise
$r_v(m)$ is 1 if $m=1$, -1 if $m=3$ and 0 otherwise
$u_j(z)$ and $v_j(z)$ are the real and imaginary parts of $f_j(z)$.

 

[benorin]

What does the notation $s\in\left\{ 0,1\right\}^k$ mean?

 

[andrewkirk]

It means the sum is for s ranging over all elements of the set $\{0,1\}^k$, which is the set of all sequences of $k$ numerals each of which is either 0, or 1. The set contains $2^k$ different sequences. $s_j$ denotes the $j$th numeral in sequence $s$.

 

[zinq]

Assuming the infinite product converges: If you can write each factor z_n of the infinite product as z_n = r_n e^iθ_n (where of course r_n > 0 and θ_n ∈ R), then you know that Re(product) = r cos(θ) and Im(product) = r sin(θ), where r = ∏ r_n and θ = ∑ θ_n.

 

[benorin]

I take a particular interest in the Gamma function, and I'm curious why I've never read any formulas for the real and imaginary parts of the Gamma function in any of my reading. Granted I have only read things in library books and websites and the texts I personally own, I do not have access to pay maths journals and such, but wouldn't these be useful to someone? I personally cannot get WolframAlpha to output these formulae as if either I do not know how to ask it for them or it does not know them (I did pay for it as a phone app, idk if that matters but I imagine they would need to make a business decision as to what constitutes premium content so perhaps it does matter but only thing I've noticed is that certain sep-by-step solutions you need to pay for). I've attempted to write out expressions for $\Re \left[ \Gamma (x+iy)\right]$ and $\Im \left[\Gamma (x+iy)\right]$ but ran into a few things I did not know how to deal with yet, still learning I guess. Is it that expressions for these are known but not particularly useful? I searched NIST's website as well.