I want to find the local minimum in the region 0.5<x<=1
of a function which has an unassigned variable r
(note: r
is always a positive integer).
I can make FindMinimum
work, but only provided I assign a specific value to r
. For example:
FindMinimum[(1 + Csc[(Pi*x)/(1 + r)]*Sin[(Pi*(1 + 2*r)*x)/(1 + r)])/
(2 + 2*r) /. r -> 2, {x, 0.5}]
But how do I find the minimum in general terms, i.e., in the form {x -> f[r]}
for some function f
? Or is the issue that there isn't a systematic representation of this local minimum with changing r
?
Note that, at least by eye, the local minimum of the original function does seem to converge for large r
:
Plot[
{
(1 + Csc[(Pi*x)/(1 + r)]*Sin[(Pi*(1 + 2 r)*x)/(1 + r)])/(2 + 2 r)/. r -> 1,
(1 + Csc[(Pi*x)/(1 + r)]*Sin[(Pi*(1 + 2 r)*x)/(1 + r)])/(2 + 2 r)/. r -> 10,
(1 + Csc[(Pi*x)/(1 + r)]*Sin[(Pi*(1 + 2 r)*x)/(1 + r)])/(2 + 2 r)/. r -> 100,
(1 + Csc[(Pi*x)/(1 + r)]*Sin[(Pi*(1 + 2 r)*x)/(1 + r)])/(2 + 2 r)/. r -> 1000,
(1 + Csc[(Pi*x)/(1 + r)]*Sin[(Pi*(1 + 2 r)*x)/(1 + r)])/(2 + 2 r)/. r -> 10000
},
{x, 0.5, 1.05},
GridLines -> Automatic,
PlotRange -> {{0.5, 1.05}, {-0.25, 0.05}},
PlotLegends -> {"r=1", "r=10", "r=100", "r=1,000", "r=10,000"}
]
EDIT:
To clarify, what I am really hoping for is:
- To define the point on the
x
axis which the local minimum converges towards with increasingr
- To find a definition in terms of
x
(defined at the point of convergence) andr
of the value of the original function with increasingr
- does the value of the minimum converge on a limit? And can it be defined algebraically?