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21. Let 12 < ˛ Ä 2. 2 . e s/ sin. 27) ; Res D 1 ; 2 ˛i . 22. 0; 1/ ! e. 0; 1/: . , [Wid46, p. 23 ([Poll48]). The Mittag-Leffler function of negative argument E˛ . x/ is completely monotonic for all 0 Ä ˛ Ä 1. G Since E0 . 1 C x/ and E1 . x/ D e x there is nothing to be proved in these cases. Let 0 < a < 1. By a standard representation [Bie31] 1 E˛ . tan / x from x D C1 to x D %, % > 0; C2 : an arc of the circle jzj D % sec , Ä argz Ä ; C3 : the reflection of C1 in the x-axis. We assume > =˛ > =2 while Z % is arbitrary but fixed.

1 in various sectors of the complex plane. These properties can be summarized as follows. 6. 1 ˛k/ kD1 1 p ; jzj ! 1 ˛k/ kD1 ; jzj ! 1 ˛k/ kD1 1 p jzj ! 17) and C inclusively. and where argz can take any value between G 10 . 14). 18) Substituting the expansion 1 z D p X kD1 k 1 zk C p zp . e. has to be omitted. 15) also hold for p D 0. 18). 22). 20 . 2) with " D 1. 1Iˇ/ 1=˛ p z d z 2 G . 23) the domain Â Ä jargzj Ä is contained in the domain G . 1Iˇ/ : 0 3 . 5) is true for any ˛ > 0 and p 0. 5).

4) In the most interesting case, Re ˛ > 0, the Mittag-Leffler function is an entire function. , [Lev56]). 1// ; k ! ˛k C 1/j ! D e Á˛ : ˛ If Re ˛ > 0, and Im ˛ 6D 0, the corresponding result is valid too. ˛0 k C 1/ ˇ for certain positive constants C1 ; C2 and sufficiently large k. 1 (Order and type). For each ˛; Re ˛ > 0; the Mittag-Leffler 1 and type D 1. , [Phr04] and [GoLuRo97]). , [Buh25a]). 1 it follows that the function E˛ . ˛ z/ D 1 X kD0 has order D . ˛k C 1/ > 0; 1 and type . 2 Relations to Elementary and Special Functions The Mittag-Leffler function plays an important role among special functions.