Incorrect answer zero where correct answer is very large (precision issue)

Question

Define a function F as follows:

F[α_, x_] = 
  N[Exp[x/2] Abs[Gamma[α/2 + I x /(2 Pi)]]^2, 10];

It is a product of two terms. As $x\to\infty$, the first term exponentially increases while the second term exponentially decreases. Their product moderately increases by a power-law.

If I evaluate F[6, 10000], the answer is 6.4163*10^16, which seems reasonable. However, if I evaluate F[6, 10000.1], the answer is 0. This is not expected since F is a continuous function.

I think the issue is on the precision, so I modified the definition F by removing N.

F[α_, x_] = 
  Exp[x/2] Abs[Gamma[α/2 + I x /(2 Pi)]]^2;

Then, even Gfintemp[6, 10000] // N gives the wrong answer 0..

How can I fix the problem, so that F gives a reasonable output for moderate values of $x$, written in exact number or numeric?

Answer 1

Try higher precision input than machine precision. Set the precision inside the function if you want to handle machine-precision input. You can use SetPrecision[] or Rationalize[]. Rationalize[] tends to change the value of the input slightly but also produce smaller denominators.

f[\[Alpha]_, x_] := 
  With[{\[Alpha]0 = SetPrecision[\[Alpha], Infinity], 
    x0 = SetPrecision[x, Infinity]},
   N[Exp[x0/2] Abs[Gamma[\[Alpha]0/2 + I x0/(2 Pi)]]^2, 10]
   ];

f2[\[Alpha]_, x_] := 
  With[{\[Alpha]0 = Rationalize[Rationalize@\[Alpha], 0], 
    x0 = Rationalize[Rationalize@x, 0]},
   N[Exp[x0/2] Abs[Gamma[\[Alpha]0/2 + I x0/(2 Pi)]]^2, 10]
   ];

Evaluation is nonzero:

f[6, 10000.1]

(*  6.416572393*10^16  *)

Relative error between the methods on this expression is negligible at 10-digit precision:

(f2[6, 10000.1] - f[6, 10000.1])/f[6, 10000.1]

(*  0.*10^-10  *)

The difference in the input:

With[{x = 10000.1}, 
 N[Rationalize[Rationalize@x, 0] - SetPrecision[x, Infinity], 10]]

(*  -3.637978807*10^-13  *)