Fig. 1

Contour plot of the function \(f_{\nu }(\lambda )\) in the complex \(\lambda \)-plane. a On the blue lines, the real part of the function \(f_{\nu }(\lambda )\) vanishes, i.e. \(\text {Re}(f_{\nu }(\lambda ))=0\), and, on the red lines, the imaginary part of the function \(f_{\nu }(\lambda )\) vanishes, i.e. \(\text {Im}(f_{\nu }(\lambda ))=0\). The eigenvalues are located at the intersection points of the blue lines and red lines. For a real index (shown for \(\nu =2\) and \(\gamma = 5\) as an example), all intersection points are located on the real axis. b For a complex index (examplarily for \(\nu =2+\mathrm {i}\) and \(\gamma =5\)), the contour plots become more sophisticated and the intersection points are displaced into the complex plane.