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3 Results

[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 , .

3.1 Complex Zeros

[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]

Theorem 3.1.

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

Theorem 3.2.

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.

3.2 Contour Lines

[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 .