Gamma Function: Visualization for Complex Arguments

Source: Hacker News

Dynamic Visualization of the Complex Gamma Graph

A real‑time exploration of the complex gamma function in motion.
Watch how the real and imaginary components evolve as the parameter shifts, creating a continuous sweep through the complex plane—mathematics unfolding like a film.

Behavior of the Complex Gamma Function

The figure shows the Euler gamma function for a complex argument of the form (\Gamma(x + i,c)).

• When the application starts, the imaginary part is set to (c = 0). In this case the graph corresponds to the classical gamma function for real arguments (\Gamma(x)).
• As (c) increases, the argument acquires an imaginary component and the gamma function takes complex values. Two curves are displayed:
  • Blue curve – the real part (\operatorname{Re}\bigl(\Gamma(x + i,c)\bigr))
  • Purple curve – the imaginary part (\operatorname{Im}\bigl(\Gamma(x + i,c)\bigr))

Example: To evaluate (\Gamma(-2.3 + 0.18i)), locate (x = -2.3) on the horizontal axis. The blue value at that position gives the real part of the result, and the purple value gives the imaginary part.

When (c) becomes sufficiently large that further variation no longer provides additional structural insight, (c) is reset to zero and then allowed to decrease into the negative direction. The negative sweep mirrors the imaginary part with respect to the (x)-axis while leaving the real part unchanged. After returning to (c = 0), the process repeats from the initial configuration.

Application Background

The program vanilla‑gamma‑graph originates from the domain www.zeta-calculator.com. At the core of that site are two JavaScript functions:

1. vanilla_zeta()
2. vanilla_gamma()

For this application only vanilla_gamma() is needed; it computes the result of complex gamma functions. The function was derived from a modification of vanilla_zeta(), which was originally developed to compute values of the Riemann zeta function. By giving vanilla_gamma() a practical use, the graphical behavior of the Euler gamma function becomes clearly visible.

Additional Information

• The vanilla_gamma() function can be copied directly from www.zeta-calculator.com.
• It is released under the Creative Commons Zero v1.0 Universal license, meaning you are free to use it in any way you like.
• Detailed instructions on how to apply vanilla_gamma() are also available on the site.

Compared to other gamma calculators, vanilla_gamma() offers a straightforward implementation with few comparable standalone alternatives. The recommendation is simple: try it—its design lets you get started quickly without spending much time on setup.

Controls

• Press: X icon to resume the gamma simulation.
• Press: ζ(s) icon to visit www.zeta-calculator.com.