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# 2 Method of Calculation

[page 639, §1]

## 2.1 Recursion Relation

[639.1.1] In the rest of this paper and are real numbers with . [639.1.2] Calculating for the case can be reduced to the case by virtue of the recursion relation [2, 21]

 (6)

valid for all , where . [639.1.3] Here denotes the largest integer smaller than . [639.1.4] Because of the recursion we consider only the case in the following.

## 2.2 Regions

[639.2.1] For the complex -plane is partitioned into four regions , , , . [639.2.2] In each region a different method is used for the calculation of the Mittag-Leffler function. [639.2.3] The central region

 (7)

is the closure of the open disk of radius centered at the origin. [639.2.4] In our numerical calculations we have chosen throughout. [639.2.5] The regions are defined by

 (8) (9)

where

 (10)

is the open wedge with opening angle . [639.2.6] For future reference we define also the (left) wedge

 (11)

[639.2.7] Finally the region is defined as

 (12)

where will be defined below.

## 2.3 Taylor series

[639.3.1] For , i.e. for the central disk region (with ), the Mittag-Leffler function is computed from truncating its Taylor series (1). [639.3.2] For given and accuracy we choose [page 640, §0]    such that

 (13)

holds. [640.0.1] The dependence of on the accuracy and other parameters is found as

 (14)

## 2.4 Integral Representations

[640.1.1] We start from the basic integral representation [2, p. 210]

 (15)

where the path of integration in the complex plane starts and ends at and encircles the circular disc in the positive sense. [640.1.2] When this integral is evaluated several cases arise. [640.1.3] For we distinguish the cases and and compute the Mittag-Leffler function from the integral representations [22]

 (16) (17)

where

 (18)
 (19)
 (20)
 (21)

[640.1.4] For the integral representations read

 (22) (23)

where the integrands have been defined in eq. (18)–(21) above.

[page 641, §1]
[641.1.1] The integrand is oscillatory but bounded over the integration interval. [641.1.2] Thus the integrals over can be evaluated numerically using any appropriate quadrature formula. [641.1.3] We use a robust Gauss-Lobatto quadrature.

[641.2.1] The integrals over involve infinite intervals. [641.2.2] For given (resp. ) and accuracy we approximate the integrals by truncation. [641.2.3] The error

 (24)

depends on the truncation point . [641.2.4] For we find

 (25)

while for we have

 (26)

## 2.5 Exponential Asymptotics

[641.3.1] The asymptotic expansions given in eqs. (21) and (22) on page 210 in Ref. [2] indicate that the Mittag-Leffler function exhibits a Stokes phenomenon. [641.3.2] The Stokes lines are the rays and it was recently shown in [23] that a Berry-type smoothing applies. [641.3.3] The exponentially improved asymptotic expansions will be used here for computing for . [641.3.4] More precisely for we use the following exponentially improved uniform asymptotic expansions [23] :

 (27)

for

 (28)

for ,

 (29)

[page 642, §0]    for with and , and

 (30)

for with and . [642.0.1] Here is a small number, in practice we use . [642.0.2] In eqs. (2.5) and (2.5)

 (31)

denotes the complex complementary error function. [642.0.3] Note the difference between eqs. (27)–(2.5) and the expansions in [2, p. 210]. [642.0.4] We take

 (32)

to truncate the asymptotic series. [642.0.5] Choosing for the machine precision we fix the radius in at

 (33)

[642.0.6] The remainder estimates in eqs. (27)–(2.5) are only valid for a value of that is chosen in an optimal sense as in eqs. (32) and (33).