Roots of Functions:
\(
\quad F(z) = \displaystyle \sum_{j = 0}^n \alpha_j \; f_j(z)
\)
where
\( \alpha_j \in \{ -1, +1 \} \)
Scroll Down For Some Screen Shots

\( f_j(z) \) =
z^j
z^j/j!
z^j/sqrt(j!)
sqrt(n choose j) z^j
cos(jz)
sin(jz)
1, sin(z), cos(z), sin(2z), cos(2z),...
1, i sin(z), cos(z), i sin(2z), cos(2z),...
exp(ijz)
cosh(jz)
Chebyshev 1st kind
Chebyshev 2nd kind
Legendre
,
n = ,
\( \alpha_i \)'s =
(-1,1)
(-1,0,1)

Size =
x center =
y center =

Renormalize intensity for real and complex roots together?
Yes
No

Compute

This app computes and plots all roots of functions of the form

\(
F(z) = \displaystyle \sum_{j = 0}^n \alpha_j \; f_j(z)
\)
for which all the coefficients \( \alpha_j \)'s are either \(+1\) or \(-1\)
and the functions \( f_j(z) \) are arbitrary analytic functions on
the complex plane.
The functions \( f_j(z) \) can be selected using the pull-down menu above.
The default choice of \( f_j(z) = z^j \) gives us \(n\)-th degree polynomials.
The number of terms, \(n\), can also be changed.

Here's my Matlab version of the code:
http://www.princeton.edu/~rvdb/WebGL/PlusMinusOne.m

Here's a related webpage by Dan Christensen:
http://jdc.math.uwo.ca/roots/
Here's a related webpage by John Baez:
http://math.ucr.edu/home/baez/roots/

Here's a screenshot for the \(18\)-degree polynomial case...

Here's a screenshot for a closeup view of the \(18\)-degree polynomial case...

Here's a screenshot for a closeup view of the \(16\)-degree polynomial case...

Here's a screenshot for the \(12\)-degree polynomial case with +1,0,-1
coefficients...

Here's a screenshot for a closeup view of the \(12\)-degree polynomial case with +1,0,-1
coefficients...
Updated 2017 Oct 25