English: The Gamma function analytically continues the factorial function to non-integers. However it is not unique in this.

Here a periodic function which is 0 at the integer values shows how it may be added to the gamma function resulting in another analytic continuation of the factorials.

Generated with sympy with jupyter.

g = \\int_{0}^{\\infty} x^{z - 1} e^{- x}\\, dx

h = \\sin{\\left (\\pi z \\right )} + \\int_{0}^{\\infty} x^{z - 1} e^{- x}\\, dx

plot(h, g, (z, 1, 4), axis_center=(1,1), ylabel="", xlabel="")

Sage version plotted with

plot([gamma(x), gamma(x)+sin(pi*x)], x,-5, 5, ymax=5, ymin = -1, detect_poles=True, gridlines=True, ticks_integer=True)