Dynamic Visualization of the Complex Gamma Graph.
A real-time exploration of the complex gamma function in motion. Watch how real and imaginary components evolve as the parameter shifts. A continuous sweep through the complex plane — mathematics unfolding like a film.
The figure shows the behavior of the Euler gamma function for a complex argument of the form Γ(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 Γ(x).
As c increases, the argument acquires an imaginary component and the gamma function takes complex values. Two curves are displayed: the blue curve represents the real part Re(Γ(x + i·c)), and the purple curve represents the imaginary part Im(Γ(x + i·c)). For example, to evaluate Γ(-2.3 + 0.18i), locate x = -2.3 on the horizontal axis. At this position, the blue value represents the real part of the result, and the purple value represents the imaginary part of the result.
If the parameter c becomes sufficiently large so that further variation no longer provides additional structural insight, c is reset to zero and then allowed to decrease into the negative direction. The behavior observed during the negative sweep is essentially identical to that of the positive sweep, except that the imaginary part is mirrored with respect to the x-axis, while the real part remains unchanged. After returning to c = 0, the entire process begins again from the initial configuration.
Background to this application:
The program "vanilla-gamma-graph" has its roots in the domain www.zeta-calculator.com. At the core of that site are two JavaScript functions: 1. vanilla_zeta() and 2. vanilla_gamma(). For this application, we only need vanilla_gamma(), which computes the result of complex gamma functions.
The function vanilla_gamma() originated from a modification of vanilla_zeta(), which was originally developed to compute values of the Riemann zeta function. The program "vanilla-gamma-graph" was created from the idea of giving the vanilla_gamma() function a practical use. What emerged is a clear insight into the graphical behavior of the Euler gamma function. Enjoy exploring it.
In addition: The vanilla_gamma() function can easily be copied and pasted from www.zeta-calculator.com. Once you possess the vanilla_gamma() function, 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 zeta-calculator.com.
What speaks in favor of vanilla_gamma() compared to other gamma calculators? I could write at length about this; briefly stated, vanilla_gamma() has advantages and disadvantages compared to other implementations. In addition, there are relatively few comparable standalone implementations available, so the field of direct comparison remains limited. My recommendation is simple: try it. The function is designed in a straightforward way, so you will not lose much time working with it.
-Press: X icon to resume the gamma simulation.
-Press: ζ(s) icon to visit www.zeta-calculator.com