paratusojitos0yx

2022-11-17

Doing some computations and plottings I've found out that the function

${}_{2}{F}_{1}(2s-1,s-{\textstyle \frac{1}{2}};s;-1)$

behaves for large real s like ${4}^{-s}$. More precisely: It seems that

${}_{2}{F}_{1}(2s-1,s-{\textstyle \frac{1}{2}};s;-1){4}^{s}$

grows, but very very slowly. No exponential growth or decay at all. If you modify the 4 only slightly this of course changes and you get exponential groth or decay. So my question is to explain this phenomenon. Is

$\underset{s\to \mathrm{\infty}}{lim}{\text{}}_{2}{F}_{1}(2s-1,s-{\textstyle \frac{1}{2}};s;-1){4}^{s}=\mathrm{\infty}?$

Is there an easy to detemine for every x>0 the respective b(x)>0 such that you have

${}_{2}{F}_{1}(2s-1,s-{\textstyle \frac{1}{2}};s;-x)\sim {b}^{-s}?$

What is

$\underset{s\to \mathrm{\infty}}{lim}{\text{}}_{2}{F}_{1}(2s-1,s-{\textstyle \frac{1}{2}};s;-x){b}^{s}$

then?

${}_{2}{F}_{1}(2s-1,s-{\textstyle \frac{1}{2}};s;-1)$

behaves for large real s like ${4}^{-s}$. More precisely: It seems that

${}_{2}{F}_{1}(2s-1,s-{\textstyle \frac{1}{2}};s;-1){4}^{s}$

grows, but very very slowly. No exponential growth or decay at all. If you modify the 4 only slightly this of course changes and you get exponential groth or decay. So my question is to explain this phenomenon. Is

$\underset{s\to \mathrm{\infty}}{lim}{\text{}}_{2}{F}_{1}(2s-1,s-{\textstyle \frac{1}{2}};s;-1){4}^{s}=\mathrm{\infty}?$

Is there an easy to detemine for every x>0 the respective b(x)>0 such that you have

${}_{2}{F}_{1}(2s-1,s-{\textstyle \frac{1}{2}};s;-x)\sim {b}^{-s}?$

What is

$\underset{s\to \mathrm{\infty}}{lim}{\text{}}_{2}{F}_{1}(2s-1,s-{\textstyle \frac{1}{2}};s;-x){b}^{s}$

then?

siriceboynu1

Beginner2022-11-18Added 12 answers

$f(s)={\phantom{\rule{thinmathspace}{0ex}}}_{2}{F}_{1}(2s-1,s-\frac{1}{2};s;-1)={\phantom{\rule{thinmathspace}{0ex}}}_{2}{F}_{1}(s-\frac{1}{2},2s-1;s;-1)$

Even for small values of s, we have a nice logarithmic behaviour

$\mathrm{log}[f(s)]\sim a-b\phantom{\rule{thinmathspace}{0ex}}s$

Using the data computed for $1\le n\le 100$, we have, with ${R}^{2}=0.9999982$,

$\begin{array}{clcl}& \text{Estimate}& \text{Standard Error}& \text{Confidence Interval}\\ a& 1.640373& 0.021344& \{1.598011,1.682736\}\\ b& 1.379308& 0.000367& \{1.380037,1.378580\}\end{array}$

and, as you noticed, b is quite close to $\mathrm{log}(4)=1.38629$

Pushing the numerical analysis much further, b is closer and closer to $(\mathrm{log}(4)-\u03f5)$. This is normal since

${\phantom{\rule{thinmathspace}{0ex}}}_{2}{F}_{1}(2s,s;s;-1)={4}^{-s}$

What is interesting is that, if $\mathrm{log}[f(s)]$ is an increasing function going to infinity while, if $\mathrm{log}[f(s)]$ goes through a maximum value.

What is interesting is that

${4}^{s}{\phantom{\rule{thinmathspace}{0ex}}}_{2}{F}_{1}(s-1,2s-1;s;-1)=2+2\sqrt{\pi}\frac{\mathrm{\Gamma}(s)}{\mathrm{\Gamma}(s-\frac{1}{2})}$

Even for small values of s, we have a nice logarithmic behaviour

$\mathrm{log}[f(s)]\sim a-b\phantom{\rule{thinmathspace}{0ex}}s$

Using the data computed for $1\le n\le 100$, we have, with ${R}^{2}=0.9999982$,

$\begin{array}{clcl}& \text{Estimate}& \text{Standard Error}& \text{Confidence Interval}\\ a& 1.640373& 0.021344& \{1.598011,1.682736\}\\ b& 1.379308& 0.000367& \{1.380037,1.378580\}\end{array}$

and, as you noticed, b is quite close to $\mathrm{log}(4)=1.38629$

Pushing the numerical analysis much further, b is closer and closer to $(\mathrm{log}(4)-\u03f5)$. This is normal since

${\phantom{\rule{thinmathspace}{0ex}}}_{2}{F}_{1}(2s,s;s;-1)={4}^{-s}$

What is interesting is that, if $\mathrm{log}[f(s)]$ is an increasing function going to infinity while, if $\mathrm{log}[f(s)]$ goes through a maximum value.

What is interesting is that

${4}^{s}{\phantom{\rule{thinmathspace}{0ex}}}_{2}{F}_{1}(s-1,2s-1;s;-1)=2+2\sqrt{\pi}\frac{\mathrm{\Gamma}(s)}{\mathrm{\Gamma}(s-\frac{1}{2})}$

Find the volume V of the described solid S

A cap of a sphere with radius r and height h.

V=??

Whether each of these functions is a bijection from R to R.

a) $f(x)=-3x+4$

b) $f\left(x\right)=-3{x}^{2}+7$

c) $f(x)=\frac{x+1}{x+2}$

?

$d)f\left(x\right)={x}^{5}+1$In how many different orders can five runners finish a race if no ties are allowed???

State which of the following are linear functions?

a.$f(x)=3$

b.$g(x)=5-2x$

c.$h\left(x\right)=\frac{2}{x}+3$

d.$t(x)=5(x-2)$ Three ounces of cinnamon costs $2.40. If there are 16 ounces in 1 pound, how much does cinnamon cost per pound?

A square is also a

A)Rhombus;

B)Parallelogram;

C)Kite;

D)none of theseWhat is the order of the numbers from least to greatest.

$A=1.5\times {10}^{3}$,

$B=1.4\times {10}^{-1}$,

$C=2\times {10}^{3}$,

$D=1.4\times {10}^{-2}$Write the numerical value of $1.75\times {10}^{-3}$

Solve for y. 2y - 3 = 9

A)5;

B)4;

C)6;

D)3How to graph $y=\frac{1}{2}x-1$?

How to graph $y=2x+1$ using a table?

simplify $\sqrt{257}$

How to find the vertex of the parabola by completing the square ${x}^{2}-6x+8=y$?

There are 60 minutes in an hour. How many minutes are there in a day (24 hours)?

Write 18 thousand in scientific notation.