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4 Inverse Mittag-Leffler functions

[647.7.1] We introduce the inverse generalized Mittag-Leffler functions \mbox{\rm L}_{{\alpha,\beta}}(z) as the solutions of the equation

\mbox{\rm L}_{{\alpha,\beta}}(\mbox{\rm E}_{{\alpha,\beta}}(z))=z. (43)

[647.7.2] Our ability to calculate \mbox{\rm E}_{{\alpha,\beta}}(z) allows us to evaluate also \mbox{\rm L}_{{\alpha,\beta}}(z) by solving this functional equation numerically. [647.7.3] We have succeeded to determine the principal branch of \mbox{\rm L}_{{\alpha,\beta}}(z) in such a way that three conditions are fulfilled. [647.7.4] 1. The function \mbox{\rm L}_{{\alpha,\beta}}(z) is single valued and well defined on its principal branch. [647.7.5] 2. Its principal branch reduces to the principal branch of the logarithm for \alpha\to 1. [647.7.6] 3. Its principal branch is a simply connected subset of the complex plane. [647.7.7] Figure 15 shows the principal branch for the case \alpha=0.95, \beta=1.

Figure 15: The dark region corresponds to the principal branch of the inverse generalized Mittag-Leffler function \mbox{\rm L}_{{\alpha,\beta}}(z) for \alpha=0.95, \beta=1.