3 Results

[642.2.1] Contrary to the exponential function the Mittag-Leffler functions exhibit complex zeros denoted as z0. [642.2.2] The complex zeros were studied by Wiman [8] who found the asymptotic curve along which the zeros are located for 0<α<2 and showed that they fall on the negative real axis for all α≥2. [642.2.3] For real α,β these zeros come in complex conjugate pairs. [642.2.4] The pairs are denoted as zk0⁢α with integers k∈Z where k>0 (resp. k≤0) labels zeros in the upper (resp. lower) half plane. [642.2.5] Figure 1 shows lines that the complex zeros zk0⁢α, k=-5,…,6 of Eα,1⁢z trace out as functions of α for 0.1≤α≤0.99995. [642.2.6] Figure 1 gives strong numerical evidence that the distance between zeros diminishes as α→0. [642.2.7] Moreover all zeros approach the point z=1 as α→0. [642.2.8] This fact seems to have been overlooked until now. [642.2.9] Of course, for every fixed α>0 the point z=1 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]

[643.0.1] For the proof we note that Wiman showed [8, p. 226]

and further that the number of zeros with zk0<r is given by r1/α/π-1+α/2 [8, p. 228]. [643.0.2] From this follows that

for all k∈Z. [643.0.3] Taking the limit α→0 in (35) and (36) gives zk⁢0=1 and arg⁡zk⁢0=0 for all k∈Z.

[page 644, §1]   
[644.1.1] The theorem states that in the limit α→0 all zeros collapse into a singularity at z=1. [644.1.2] Next we turn to the limit α→1. [644.1.3] Of course for α=1 we have E1,1⁢z=exp⁡z which is free from zeros. [644.1.4] Figure 1 shows that this is indeed the case because, as α→1, the zeros approach -∞ along straight lines parallel to the negative real axis. [644.1.5] In fact we find

[644.1.6] The theorem shows that the phase switches as α crosses the value α=1 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 α for the case 1<α<2 is illustrated in Figure 2. [644.2.2] Note that with increasing α 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 α=2 the zeros -k-1/22⁢π2 all fall on the negative real axis as can be seen in Figure 11 below. [644.2.7] For α>2 all zeros lie on the negative real axis.

[644.3.1] Next we present contour plots for Re⁡Eα,β⁢z. [644.3.2] We use the notation

for the contour lines of the real and imaginary part. [644.3.3] The region z∈C:Re⁡Eα,β⁢z>1 will be coloured white. [644.3.4] The region z∈C:Re⁡Eα,β⁢z<-1 will be coloured black. [644.3.5] The region z∈C:0≤Re⁡Eα,β⁢z≤1 is light gray. [644.3.6] The region z∈C:-1≤Re⁡Eα,β⁢z≤0 is dark gray. [644.3.7] Thus the contour line Cα,βRe⁢0 separates the light gray from dark gray, the contour Cα,βRe⁢1 separates white from light gray, and Cα,βRe⁢-1 dark gray from black. [644.3.8] Because Re⁡Eα,β⁢z 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 α→0. Setting α=0 the series (1) defines the function

for all z<1. [645.1.2] This function is not entire, but can be analytically continued to all of C∖1, and has then a simple pole at z=1 for all β. [645.1.3] In Figure 3 we show the contour plot for the case α=0, β=1. [645.1.4] The contour line C0,1Re⁢0 is the straight line Re⁡z=1 separating the left and right half plane. [645.1.5] The contour line C0,1Re⁢1 is the boundary circle of the white disc on the left, while the contour line C0,1Re⁢-1 is the boundary of the black disc on the right.

[645.2.1] Having discussed the case α=0 we turn to the case α>0 and note that the limit α→0 is not continuous. [645.2.2] For α>0 the Mittag-Leffler function is an entire function. [645.2.3] As an example we show the contour plot for α=0.2, β=1 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 α between 0 and 0.2. [646.0.2] It seems as if the singularity at z=1 for α=0 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 α⁢π/2.

[646.1.1] The behaviour of Eα,β⁢z for 0<α<2 is generally dominated by the wedge W+⁢α⁢π/2 indicated by dashed lines in Figure 4. [646.1.2] For z∈W+⁢α⁢π/2 the Mittag-Leffler function grows to infinity as z→∞. [646.1.3] Inside this wedge the function oscillates as a function of Im⁡z. [646.1.4] For z∈W-⁢α⁢π/2 the function decays to zero as z→∞. [646.1.5] Along the delimiting rays, i.e. for arg⁡z=±α⁢π/2, the function approaches 1/α 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 Re⁡Eα,β⁢z is symmetric with respect to the real axis.

[646.3.1] Contrary to C0,1Re⁢0 the contour line C0.2,1Re⁢0 consists of infinitely many pieces. [646.3.2] These pieces will be denoted as C0.2,1Re⁢0;±k with k=1,2,3,… located in the upper (+) resp. lower (-) half plane. [646.3.3] The numbering is chosen from left to right, so that C0.2,1Re⁢0;±1 separates the light gray region in the left half plane from the dark gray in the right half plane. [646.3.4] The line C0.2,1Re⁢0;+2 is the boundary of the light gray region surrounding the first white “finger” (lobe) in the upper half plane and C0.2,1Re⁢0;-2 is its reflection on the real axis. [646.3.5] Similarly for k=3,4,…. [646.3.6] Note that C0.2,1Re⁢0;±1 seem to encircle the central white lobe (“bubble”) by going to ±i⁢∞ parallel to the imaginary axis.

[646.4.1] With increasing α the wedge W+⁢α⁢π/2 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 Cα,1Re⁢0;±1 moves to the left. [646.4.3] This is illustrated in a threedimensional plot of Re⁡E0.333,1⁢z in Figure 5. [646.4.4] In this Figure we have indicated also the complex zeros as the intersection of C0.333,1Re⁢0 (shown as thick solid lines) and C0.333,1Im⁢0 (shown as thick dashed lines). [646.4.5] At α=1/2 the contours C0.5,1Re⁢0;±1 cross the imaginary axis.

[page 647, §1]   
[647.1.1] Around α≈0.73 the contours Cα,1Re⁢0;±2 osculate the contours Cα,1Re⁢0;±1. [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 Cα,1Re⁢0;±1 and Cα,1Re⁢0;±k for k=2,3,4,…. [647.1.5] We estimate the value of α for the first osculation at α≈0.734375±0.000015.

[647.2.1] For α→1 the dark gray fingers (where Re⁡Eα,β<0) extend more and more into the left half plane. [647.2.2] For α=1 the wedge W+⁢π/2 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 C1,1Re⁢0;±k degenerate into

i.e. into straight lines parallel to the real axis. [647.2.5] This case is shown in Figure 7. [647.2.6] For α=1+ϵ with ϵ>0 the gray fingers are again finite. [647.2.7] This is shown in Figure 8.

[647.3.1] As α is increased further the fingers grow thicker and approach each other near the negative real axis. [647.3.2] For α≈1.42215±0.00005 the first of an infinite cascade of osculations appears. [647.3.3] This is shown in Figure 9. [647.3.4] The limit α→2 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 α>2 the behaviour changes drastically. [647.4.2] Figure 12 shows the contour plot for α=3, β=1. [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 α<2. [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 β≠1 choosing α=1/3 for illustration. [647.5.2] For β>1 equation (41) implies that the central lobe first grows (because Γ⁢β diminishes) and then shrinks as β→∞ for small values of α. [647.5.3] This is illustrated in the upper left subfigure of Figure 13. [647.5.4] For reference the case β=1 is also shown in the upper right subfigure of Figure 13.

[647.6.1] More interesting behaviour is obtained for β<1. [647.6.2] In this case the contours C1/3,βRe⁢0;±1 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 C1/3,βRe⁢0;±k and C1/3,βRe⁢0;±k+1 begins starting from k=∞. [647.6.5] One of the last of these osculations can be seen for β=1/3 on the lower right subfigure of Figure 13. [647.6.6] As β falls below 1/3 the contour C1/3,βRe⁢0;+1 coalesces with C1/3,βRe⁢0;-1 to form a new large finite central lobe. [647.6.7] This new second lobe becomes smaller and retracts towards the origin for β→0. [647.6.8] This can be seen from the upper left subfigure of Figure 14 where the case β=0 is shown. [647.6.9] As β 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 β crosses a negative integer and is a consequence of the poles in Γ⁢β.