DLMF: Sidebar 5.SB1: Gamma & Digamma Phase Plots

Charles Clark

The color encoded phases of $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ (above) and $\mathop{\psi\/}\nolimits\!\left(z\right)$ (below), are constrasted in the negative half of the complex plane.

In the upper half of the image, the poles of $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ are clearly visible at negative integer values of $z$ : the phase changes by $2\pi$ around each pole, showing a full revolution of the color wheel. This pattern is analogous to one that would be seen in fluid flow generated by a semi-infinite line of vortices.

In the lower half of the image, the poles of $\mathop{\psi\/}\nolimits\!\left(z\right)$ (corresponding to the poles of $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ ) and the zeros between them are clear. Phase changes around the zeros are of opposite sign to those around the poles. The fluid flow analogy in this case involves a line of vortices of alternating sign of circulation, resulting in a near cancellation of flow far from the real axis.