Improve the precision of the Whittaker function?

Questions
1

mu = -0.1;
beta = -0.5;
sigma = 2.5;
a = 90;
b = 120;
m = -1/(4*beta); m1 = m - 1/2;
a1 = mu/(sigmasigmabeta)a^(-2beta);
b1 = mu/(sigmasigmabeta)b^(-2beta);
MyF[k_] := Gamma[ m - k](WhittakerW[k, m1, b1]WhittakerM[k, m1, a1] - WhittakerM[k, m1, b1]*WhittakerW[k, m1, a1]);

this code MyF[1000]=0.

But for this code:

MyF1[k_] := N[ Gamma[Rationalize[ m - k]](WhittakerM[Rationalize[k], 0, Rationalize[a1]] WhittakerW[Rationalize[k], 0, Rationalize[b1]] - WhittakerM[Rationalize[k], 0, Rationalize[b1]] WhittakerW[ Rationalize[k], 0, Rationalize[a1]]), 5]; 

MyF1[1000]=0.044810;
2

Do the computation with exact numbers:

mu = -1/10; beta = -1/2; sigma = 2 + 1/2;

and turn to floating points only at the very end:

In[7]:= N[MyF[1000], 5]

Out[7]= 0.044810

Your first computation works with machine numbers, which are not enough.

3

mu=-0.1;
beta=-0.5;
sigma=2.5;

(* Add this line to manipulate precision *) 
{mu,beta,sigma}=SetPrecision[{mu,beta,sigma},20];

a=90;
b=120;
m=-1/(4*beta);m1=m-1/2;
a1=mu/(sigma*sigma*beta)a^(-2beta);
b1=mu/(sigma*sigma*beta)b^(-2beta);
MyF[k_]:=Gamma[m-k](WhittakerW[k,m1,b1]WhittakerM[k,m1,a1]-WhittakerM[k,m1,b1]*WhittakerW[k,m1,a1]);

The result is

MyF[1000]//Chop
==> 0.0448100417591471632

Link to documentation:

4

I want to find all roots of the function MyF[k]. and do you know how to do it?

5

Try FindRoot: Search for a numerical root of a function—Wolfram Documentation with WorkingPrecision -> 20