[642.1.1] In this section we present the results of extensive numerical
calculations using an algorithm that is based on the error
estimates developed above.
[642.1.2] Our results give a comprehensive picture of the behaviour
of in the complex
-plane for all values
of the parameters
,
.
[642.2.1] Contrary to the exponential function the Mittag-Leffler functions
exhibit complex zeros denoted as .
[642.2.2] The complex zeros were studied by Wiman [8] who
found the asymptotic curve along which the zeros
are located for
and showed that
they fall on the negative real axis for all
.
[642.2.3] For real
these zeros come in complex conjugate pairs.
[642.2.4] The pairs are denoted as
with integers
where
(resp.
) labels zeros in the upper (resp. lower) half plane.
[642.2.5] Figure 1 shows lines that the complex zeros
,
of
trace out as functions
of
for
.
[642.2.6] Figure 1 gives strong
numerical evidence that the distance between
zeros diminishes as
.
[642.2.7] Moreover all zeros approach the point
as
.
[642.2.8] This fact seems to have been overlooked until now.
[642.2.9] Of course, for every fixed
the point
is neither
a zero nor an accumulation point of zeros
because the zeros of an entire function must remain isolated.
[642.2.10] The numerical evidence is confirmed analytically.
[page 643, §1]
The zeros of
obey
![]() |
(34) |
for all .
[643.0.1] For the proof we note that Wiman showed [8, p. 226]
![]() |
(35) |
and further that the number
of zeros with is given by
[8, p. 228].
[643.0.2] From this follows that
![]() |
(36) |
for all .
[643.0.3] Taking the limit
in (35) and (36)
gives
and
for all
.
[page 644, §1]
[644.1.1] The theorem states that in the limit
all zeros collapse into a singularity at
.
[644.1.2] Next we turn to the limit
[644.1.3] Of course for
we have
which is free from zeros.
[644.1.4] Figure 1 shows that this is indeed
the case because, as
, the zeros approach
along straight lines parallel to the negative real axis.
[644.1.5] In fact we find
Let .
Then the zeros
of
obey
![]() |
![]() |
(37) | |
![]() |
![]() |
(38) |
for all .
[644.1.6] The theorem shows that the phase switches
as crosses the value
in the sense that minima (valleys)
and maxima (hills) of the Mittag-Leffler
function are exchanged
(see also Figures 7 and
8 below).
[644.2.1] The location of zeros as function of for the
case
is illustrated in Figure 2.
[644.2.2] Note that with increasing
more and more pairs of
zeros collapse onto the negative real axis.
[644.2.3] The collapse appears to happen in a continuous manner
(see also Figures 9 and 10
below).
[644.2.4] It is interesting to note that after two conjugate zeros
merge to become a single zero on the negative real axis
this merged zero first moves to the right towards zero and
only afterwards starts to move left towards
.
[644.2.5] This effect can also be seen in Figure 2.
[644.2.6] For
the zeros
all fall
on the negative real axis as can be seen
in Figure 11 below.
[644.2.7] For
all zeros lie on the negative real axis.
[644.3.1] Next we present contour plots for .
[644.3.2] We use the notation
![]() |
![]() |
(39) | |
![]() |
![]() |
(40) |
for the contour lines of the real and imaginary part.
[644.3.3] The region will be coloured white.
[644.3.4] The region
will be coloured black.
[644.3.5] The region
is light gray.
[644.3.6] The region
is dark gray.
[644.3.7] Thus the contour line
separates the light gray from dark
gray, the contour
separates white from light gray,
and
dark gray from black.
[644.3.8] Because
is continuous there exists
in all figures light and dark gray regions between white
and black regions even if the gray regions
cannot be discerned on a figure.
[page 645, §1]
[645.1.1] We begin our discussion with the case .
Setting
the series (1) defines the function
![]() |
(41) |
for all .
[645.1.2] This function is not entire, but can be analytically
continued to all of
, and
has then a simple pole at
for all
.
[645.1.3] In Figure 3 we show the contour plot
for the case
,
.
[645.1.4] The contour line
is the straight line
separating the left and right half plane.
[645.1.5] The contour line
is the boundary circle
of the white disc on the left, while the contour line
is the boundary of the black disc on the right.
[645.2.1] Having discussed the case we turn
to the case
and note that the limit
is not continuous.
[645.2.2] For
the Mittag-Leffler function is an entire function.
[645.2.3] As an example we show the contour plot for
,
in Figure 4.
[645.2.4] The central white circular lobe extending
[page 646, §0]
to the origin appears to be a remnant of the white disc
in Figure 3.
[646.0.1] They evolve continuously from each other
upon changing
between 0 and 0.2.
[646.0.2] It seems as if the singularity at
for
had moved along the real axis through the black
circle to
thereby producing an infinite number of
secondary white and black lobes (or fingers) confined to a
wedge shaped region with opening angle
.
[646.1.1] The behaviour of for
is generally
dominated by the wedge
indicated by dashed lines in Figure 4.
[646.1.2] For
the Mittag-Leffler
function grows to infinity as
.
[646.1.3] Inside this wedge the function oscillates as a
function of
.
[646.1.4] For
the function decays to zero as
.
[646.1.5] Along the delimiting rays, i.e. for
,
the function approaches
in an oscillatory fashion.
[646.2.1] The oscillations inside the wedge are seen as black and
white lobes (or fingers) in Figure 4.
[646.2.2] Each white finger is surrounded by a light gray region.
[646.2.3] Near the tip of the light gray region surrounding a white
finger lie complex zeros of the Mittag-Leffler function.
[646.2.4] The real part
is symmetric with respect to the real axis.
[646.3.1] Contrary to the contour line
consists of infinitely many pieces.
[646.3.2] These pieces will be denoted as
with
located in the
upper (
) resp. lower (
) half plane.
[646.3.3] The numbering is chosen from left to right, so that
separates the light gray region
in the left half plane from the dark gray in the right half plane.
[646.3.4] The line
is the boundary of the
light gray region surrounding the first white “finger”
(lobe) in the upper half plane and
is its reflection on the real axis.
[646.3.5] Similarly for
.
[646.3.6] Note that
seem to encircle the
central white lobe (“bubble”) by going to
parallel to the
imaginary axis.
[646.4.1] With increasing the wedge
opens,
the central lobe becomes smaller, the side fingers (or lobes)
grow thicker and begin to extend towards the left half plane.
[646.4.2] At the same time the contour line
moves
to the left.
[646.4.3] This is illustrated in a threedimensional plot of
in Figure 5.
[646.4.4] In this Figure we have indicated also the complex zeros
as the intersection of
(shown as thick
solid lines) and
(shown as thick
dashed lines).
[646.4.5] At
the contours
cross
the imaginary axis.
[page 647, §1]
[647.1.1] Around the contours
osculate the contours
.
[647.1.2] The osculation eliminates a light gray finger and creates
a dark gray finger.
[647.1.3] In Figure 6
we show the situation before and after the osculation.
[647.1.4] This is the first of an infinity of similar osculations
between
and
for
.
[647.1.5] We estimate the value of
for the first
osculation at
.
[647.2.1] For the dark gray fingers (where
)
extend more and more into the left half plane.
[647.2.2] For
the wedge
becomes the right half plane
and the lobes or fingers run parallel to the real axis.
[647.2.3] The dark gray fingers, and therefore the oscillations,
now extend to
.
[647.2.4] The contour lines
degenerate into
![]() |
(42) |
i.e. into straight lines parallel to the real axis.
[647.2.5] This case is shown in Figure 7.
[647.2.6] For with
the gray
fingers are again finite.
[647.2.7] This is shown in Figure 8.
[647.3.1] As is increased further the fingers grow thicker
and approach each other near the negative real axis.
[647.3.2] For
the first of an infinite
cascade of osculations appears.
[647.3.3] This is shown in Figure 9.
[647.3.4] The limit
is illustrated in
Figures 10 and 11.
[647.3.5] Note that the background colour changes from light
gray in Figure 10 to dark gray in
11 in agreement with the discussion
of complex zeros above.
[647.4.1] For the behaviour changes drastically.
[647.4.2] Figure 12 shows the contour
plot for
,
.
[647.4.3] Note the scale of the axes and hence there
are no visible dark or light gray regions.
[647.4.4] The wedge shaped region is absent.
[647.4.5] The rays delimiting the wedge may still be
viewed as if the fingers were following them
in the same way as for
.
[647.4.6] Thus the fingers are more strongly bent
as they approach the negative real axis.
[647.5.1] Now we turn to the cases choosing
for illustration.
[647.5.2] For
equation (41) implies that
the central lobe first grows (because
diminishes)
and then shrinks as
for small values of
.
[647.5.3] This is illustrated in the upper left subfigure of
Figure 13.
[647.5.4] For reference the case
is also shown
in the upper right subfigure of Figure 13.
[647.6.1] More interesting behaviour is obtained for .
[647.6.2] In this case the contours
stop to run to infinity parallel to the imaginary axis.
[647.6.3] Instead they seem to approach infinity along rays
extending into the negative half axis as illustrated
in the lower left subfigure of Figure 13.
[647.6.4] At the same time a sequence of osculations between
and
begins starting from
.
[647.6.5] One of the last of these osculations can be
seen for
on the lower right subfigure of
Figure 13.
[647.6.6] As
falls below
the contour
coalesces with
to form a new large
finite central lobe.
[647.6.7] This new second lobe becomes smaller and retracts towards
the origin for
.
[647.6.8] This can be seen from the upper left subfigure of
Figure 14 where the case
is shown.
[647.6.9] As
falls below zero the same process of formation
of a new central lobe accompanied by a cascade of
osculations starts again.
[647.6.10] This occurs iteratively whenever
crosses a negative
integer and is a consequence of the poles in
.