[page 639, §1]
[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]
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(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.
[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
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(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
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(8) | |
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(9) |
where
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(10) |
is the open wedge with opening angle .
[639.2.6] For future reference we define also the (left) wedge
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(11) |
[639.2.7] Finally the region is defined as
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(12) |
where will be defined below.
[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
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(13) |
holds.
[640.0.1] The dependence of on the accuracy
and other
parameters is found as
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(14) |
[640.1.1] We start from the basic integral representation [2, p. 210]
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(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]
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(16) | ||
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(17) |
where
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(18) |
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(19) |
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(20) |
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(21) |
[640.1.4] For the integral representations read
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(22) | ||
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(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
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(25) |
while for we have
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(26) |
[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] :
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(27) |
for
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(28) |
for ,
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||
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(29) |
[page 642, §0]
for with
and
, and
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||
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(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).