# Show that complex product does not converge

• Jun 3rd 2013, 02:36 AM
bkarpuz
Show that complex product does not converge
Dear MHF members,

I have the following product of complex numbers
$\prod_{j=1}^{\infty}\bigg[\frac{1}{2^{j}}+\mathrm{i}\bigg(1-\frac{1}{2^{j}}\bigg)\bigg]$
and Mathematica shows me that the product accumulates to 4 different points.
Each is located in different quadrant and they are outside of the circle with a radius of half.
However, I cannot compute them explicitly.

Thank you.
bkarpuz
• Jun 3rd 2013, 07:16 PM
chiro
Re: Show that complex product does not converge
Hey bkarpuz.

With regards to your original thread title (i.e. convergence), you can show that if the sum of the logs converges then the original product converges.

See if you can prove convergence first because if you can't then you won't have a unique answer.
• Jun 4th 2013, 03:16 AM
bkarpuz
Re: Show that complex product does not converge
Quote:

Originally Posted by chiro
Hey bkarpuz.

With regards to your original thread title (i.e. convergence), you can show that if the sum of the logs converges then the original product converges.

See if you can prove convergence first because if you can't then you won't have a unique answer.

Thank you very much chiro.
I can simply prove that non-convergence (actually accumulation) as follows.

Proof. As we know for any complex number $z=x+\mathrm{i}y$,
$|x|+|y|\geq|z|\geq\max\{|x|,|y|\}$.
Let $x_{n}:=\frac{1}{2^{n}}$ and $z_{n}:=x_{n}+\mathrm{i}(1-x_{n})$,
then we have $|z_{j}|\geq|1-x_{j}|$, which yields
$\bigg|\prod_{i=1}^{\infty}z_{i}\bigg|=\prod_{i=1}^ {\infty}|z_{i}|\geq\prod_{i=1}^{\infty}|1-x_{i}|\approx\ell:=0.2887880951$.
This shows that product cannot tend to $0$ (i.e., $0+\mathrm{i}0$).
On the other hand, we have
$\bigg|\prod_{i=1}^{\infty}z_{i}\bigg|\leq\prod_{i= 1}^{\infty}(|x_{i}|+|1-x_{i}|)=1$
showing that the product can not also diverge (in the sense of complex modulus).
Then, the complex sequence $w_{n}:=\prod_{i=1}^{n}z_{i}$ lies in the annulus $\mathcal{A}:\ \ell\leq|z|\leq1$.
Due to Bolzano-Weierstrass theorem (considered separetely for the real and imaginary parts),
there exists a subsequence $w_{n_{k}}$ of $w_{n}$ which converges to some $w\in\mathcal{A}$.
Recall that $\lim_{n\to\infty}z_{n}=\mathrm{i}$ and $w_{n_{k}+p}=w_{n_{k}}\prod_{i=n_{k}+1}^{p}z_{i}$,
which shows that $\lim_{k\to\infty}w_{n_{k}+p}=\mathrm{i}^{p}w$ for $p=0,1,\cdots$.
Therefore, the product accumulates to at least (also at most) four different points $\pm w,\pm\mathrm{i}w$ since $w\neq0$. $\rule{0.2cm}{0.2cm}$

But I just wanted to know if it is possible to find (by hand) what $w$ is.
• Jun 6th 2013, 04:58 AM
bkarpuz
Add: Show that complex product does not converge
Quote:

Originally Posted by bkarpuz
Proof. As we know for any complex number $z=x+\mathrm{i}y$,
$|x|+|y|\geq|z|\geq\max\{|x|,|y|\}$.
Let $x_{n}:=\frac{1}{2^{n}}$ and $z_{n}:=x_{n}+\mathrm{i}(1-x_{n})$,
then we have $|z_{j}|\geq|1-x_{j}|$, which yields
$\bigg|\prod_{i=1}^{\infty}z_{i}\bigg|=\prod_{i=1}^ {\infty}|z_{i}|\geq\prod_{i=1}^{\infty}|1-x_{i}|\approx\ell:=0.2887880951$.

Recall that
$\prod_{j=1}^{\infty}\bigg(1-\frac{\lambda^{2}}{(j\pi)^{2}}\bigg)=\frac{\sin( \lambda)}{\lambda}$.
(See Basel problem - Euler's Approach)
If $\lambda\geq\frac{3\pi}{2\sqrt{2}}$, we have $\lambda\geq\frac{j\pi}{\sqrt{2^{j}}}$ for all $j=1,2,\cdots$.
The sequence on the right-hand side increases for $j=1,2$,
reaches its maximum value $\frac{3\pi}{2\sqrt{2}}\approx3.33216$ at $j=3$,
and then decreasses for $j=4,5,\cdots$.
This yields
$1-x_{j}=1-\frac{1}{2^{j}}\geq1-\frac{\lambda^{2}}{(j\pi)^{2}}$ for all $j=1,2,\cdots$
provided that $\lambda\geq\frac{3\pi}{2\sqrt{2}}$.
Letting $\lambda\geq\frac{3\pi}{2\sqrt{2}}$ and we have for the real product
$\prod_{j=1}^{\infty}(1-x_{j})=\prod_{j=1}^{\infty}\bigg(1-\frac{1}{2^{j}}\bigg)\geq\prod_{j=1}^{\infty}\bigg (1-\frac{\lambda^{2}}{(j\pi)^{2}}\bigg)=\frac{\sin( \lambda)}{\lambda}$,
which is strictly positive for $\lambda=2\pi+\frac{1}{2}\pi>\frac{3\pi}{2\sqrt{2}}$. $\rule{0.2cm}{0.2cm}$

Quote:

Originally Posted by bkarpuz
This shows that product cannot tend to $0$ (i.e., $0+\mathrm{i}0$).
On the other hand, we have
$\bigg|\prod_{i=1}^{\infty}z_{i}\bigg|\leq\prod_{i= 1}^{\infty}(|x_{i}|+|1-x_{i}|)=1$
showing that the product can not also diverge (in the sense of complex modulus).
Then, the complex sequence $w_{n}:=\prod_{i=1}^{n}z_{i}$ lies in the annulus $\mathcal{A}:\ \ell\leq|z|\leq1$.
Due to Bolzano-Weierstrass theorem (considered separetely for the real and imaginary parts),
there exists a subsequence $w_{n_{k}}$ of $w_{n}$ which converges to some $w\in\mathcal{A}$.
Recall that $\lim_{n\to\infty}z_{n}=\mathrm{i}$ and $w_{n_{k}+p}=w_{n_{k}}\prod_{i=n_{k}+1}^{n_{k}+p}z_ {i}$,
which shows that $\lim_{k\to\infty}w_{n_{k}+p}=\mathrm{i}^{p}w$ for $p=0,1,\cdots$.
Therefore, the product accumulates to at least (also at most) four different points $\pm w,\pm\mathrm{i}w$ since $w\neq0$. $\rule{0.2cm}{0.2cm}$

But I just wanted to know if it is possible to find (by hand) what $w$ is.