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 $\mbox{\rm E}_{{\alpha,\beta}}(z)$ in the complex $z$ -plane for all values of the parameters $\alpha>0$ , $\beta\in\mathbb{R}$ .

3.1 Complex Zeros

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

Figure 1: The lines trace out the locus of complex zeros $z^{0}_{k}(\alpha)$ as functions of $\alpha$ for $k=-5,...,6$ for the the Mittag-Leffler function $\mbox{\rm E}_{{\alpha,1}}(z)$ in the range $0.1\leq\alpha\leq 0.99995$ . The line styles consecutively label $k=0,1$ (solid) $k=-1,2$ (dashed) $k=-2,3$ (dash-dotted), $k=-3,4$ (solid) $k=-4,5$ (dashed) and $k=-5,6$ (dash-dotted). The symbols mark $\alpha=0.5$ (plus), $\alpha=0.7$ (triangle right), $\alpha=0.8$ (triangle left), $\alpha=0.9$ (circle), $\alpha=0.95$ (square), $\alpha=0.99$ (asterisk), $\alpha=0.999$ (diamond), and $\alpha=0.9999$ (cross).

[page 643, §1]

Theorem 3.1.

The zeros $z^{0}_{k}(\alpha)$ of $\mbox{\rm E}_{{\alpha,1}}(z)$ obey

$\lim _{{\alpha\to 0}}z^{0}_{k}(\alpha)=1$

(34)

for all $k\in\mathbb{Z}$ .

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

$\lim _{{k\to\infty}}\arg z^{0}_{{\pm k}}(\alpha)=\pm\frac{\alpha\pi}{2}$

(35)

and further that the number of zeros with $z^{0}_{k}<r$ is given by $r^{{1/\alpha}}/\pi-1+(\alpha/2)$ [8, p. 228]. [643.0.2] From this follows that

$\left(1-\frac{\alpha}{2}\right)^{\alpha}\pi^{\alpha}<|z^{0}_{k}(\alpha)|<\left(2k+1-\frac{\alpha}{2}\right)^{\alpha}\pi^{\alpha}$

(36)

for all $k\in\mathbb{Z}$ . [643.0.3] Taking the limit $\alpha\to 0$ in (35) and (36) gives $|z_{k}(0)|=1$ and $\arg z_{k}(0)=0$ for all $k\in\mathbb{Z}$ .

[page 644, §1]
[644.1.1] The theorem states that in the limit $\alpha\to 0$ all zeros collapse into a singularity at $z=1$ . [644.1.2] Next we turn to the limit $\alpha\to 1.$ [644.1.3] Of course for $\alpha=1$ we have $\mbox{\rm E}_{{1,1}}(z)=\exp z$ which is free from zeros. [644.1.4] Figure 1 shows that this is indeed the case because, as $\alpha\to 1$ , the zeros approach $-\infty$ along straight lines parallel to the negative real axis. [644.1.5] In fact we find

Theorem 3.2.

Let $\epsilon>0$ . Then the zeros $z^{0}_{k}(\alpha)$ of $\mbox{\rm E}_{{\alpha,1}}(z)$ obey

$\displaystyle\lim _{{\epsilon\to 0}}\;\operatorname{Im}z^{0}_{k}(1-\epsilon)$	$\displaystyle=(2k-1)\pi$		(37)
$\displaystyle\lim _{{\epsilon\to 0}}\;\operatorname{Im}z^{0}_{k}(1+\epsilon)$	$\displaystyle=2k\pi$		(38)

for all $k\in\mathbb{Z}$ .

[644.1.6] The theorem shows that the phase switches as $\alpha$ crosses the value $\alpha=1$ 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 $\alpha$ for the case $1<\alpha<2$ is illustrated in Figure 2. [644.2.2] Note that with increasing $\alpha$ 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 $-\infty$ . [644.2.5] This effect can also be seen in Figure 2. [644.2.6] For $\alpha=2$ the zeros $-(k-(1/2))^{2}\pi^{2}$ all fall on the negative real axis as can be seen in Figure 11 below. [644.2.7] For $\alpha>2$ all zeros lie on the negative real axis.

Figure 2: Locus of complex zeros $z^{0}_{k}(\alpha)$ as functions of $\alpha$ for $k=\pm 1,\pm 2,\pm 3,\pm 4$ for the the Mittag-Leffler function $\mbox{\rm E}_{{\alpha,1}}(z)$ in the range $1.00001\leq\alpha\leq 1.9$ . The line styles consecutively label $k=\pm 1$ (solid) $k=\pm 2$ (dashed) $k=\pm 3$ (dash-dotted), $k=\pm 4$ (solid). The symbols mark $\alpha=1.00001$ (triangle left filled) $\alpha=1.001$ (triangle right) $\alpha=1.1$ (plus) $\alpha=1.3$ (circle) $\alpha=1.5$ (square), and $\alpha=1.7$ (diamond) $\alpha=1.9$ (cross). The arrows on the left indicate the asymptotic locations $z^{0}_{k}(1)=2k\pi$ in the limit $\alpha\to 1$ from above.

3.2 Contour Lines

[644.3.1] Next we present contour plots for $\operatorname{Re}\mbox{\rm E}_{{\alpha,\beta}}(z)$ . [644.3.2] We use the notation

$\displaystyle\mathcal{C}^{{\operatorname{Re}}}_{{\alpha,\beta}}(v)$	$\displaystyle=\{ z\in\mathbb{C}:\operatorname{Re}\mbox{\rm E}_{{\alpha,\beta}}(z)=v\}$		(39)
$\displaystyle\mathcal{C}^{{\operatorname{Im}}}_{{\alpha,\beta}}(v)$	$\displaystyle=\{ z\in\mathbb{C}:\operatorname{Im}\mbox{\rm E}_{{\alpha,\beta}}(z)=v\}$		(40)

for the contour lines of the real and imaginary part. [644.3.3] The region $\{ z\in\mathbb{C}:\operatorname{Re}\mbox{\rm E}_{{\alpha,\beta}}(z)>1\}$ will be coloured white. [644.3.4] The region $\{ z\in\mathbb{C}:\operatorname{Re}\mbox{\rm E}_{{\alpha,\beta}}(z)<-1\}$ will be coloured black. [644.3.5] The region $\{ z\in\mathbb{C}:0\leq\operatorname{Re}\mbox{\rm E}_{{\alpha,\beta}}(z)\leq 1\}$ is light gray. [644.3.6] The region $\{ z\in\mathbb{C}:-1\leq\operatorname{Re}\mbox{\rm E}_{{\alpha,\beta}}(z)\leq 0\}$ is dark gray. [644.3.7] Thus the contour line $\mathcal{C}^{{\operatorname{Re}}}_{{\alpha,\beta}}(0)$ separates the light gray from dark gray, the contour $\mathcal{C}^{{\operatorname{Re}}}_{{\alpha,\beta}}(1)$ separates white from light gray, and $\mathcal{C}^{{\operatorname{Re}}}_{{\alpha,\beta}}(-1)$ dark gray from black. [644.3.8] Because $\operatorname{Re}\mbox{\rm E}_{{\alpha,\beta}}(z)$ 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.

Figure 3: Contour plot for $\operatorname{Re}\mbox{\rm E}_{{0,1}}(z)$ . The region $\{ z\in\mathbb{C}:\operatorname{Re}\mbox{\rm E}_{{\alpha,\beta}}(z)>1\}$ is white, $\{ z\in\mathbb{C}:\operatorname{Re}\mbox{\rm E}_{{\alpha,\beta}}(z)<-1\}$ is black, $\{ z\in\mathbb{C}:0\leq\operatorname{Re}\mbox{\rm E}_{{\alpha,\beta}}(z)\leq 1\}$ is light gray, and $\{ z\in\mathbb{C}:-1\leq\operatorname{Re}\mbox{\rm E}_{{\alpha,\beta}}(z)\leq 0\}$ is dark gray.

[page 645, §1]
[645.1.1] We begin our discussion with the case $\alpha\to 0$ . Setting $\alpha=0$ the series (1) defines the function

$\mbox{\rm E}_{{0,\beta}}(z)=\frac{1}{\Gamma(\beta)(1-z)}$

(41)

for all $|z|<1$ . [645.1.2] This function is not entire, but can be analytically continued to all of $\mathbb{C}\setminus\{ 1\}$ , and has then a simple pole at $z=1$ for all $\beta$ . [645.1.3] In Figure 3 we show the contour plot for the case $\alpha=0$ , $\beta=1$ . [645.1.4] The contour line $\mathcal{C}^{{\operatorname{Re}}}_{{0,1}}(0)$ is the straight line $\operatorname{Re}z=1$ separating the left and right half plane. [645.1.5] The contour line $\mathcal{C}^{{\operatorname{Re}}}_{{0,1}}(1)$ is the boundary circle of the white disc on the left, while the contour line $\mathcal{C}^{{\operatorname{Re}}}_{{0,1}}(-1)$ is the boundary of the black disc on the right.

[645.2.1] Having discussed the case $\alpha=0$ we turn to the case $\alpha>0$ and note that the limit $\alpha\to 0$ is not continuous. [645.2.2] For $\alpha>0$ the Mittag-Leffler function is an entire function. [645.2.3] As an example we show the contour plot for $\alpha=0.2$ , $\beta=1$ 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 $\alpha$ between 0 and 0.2. [646.0.2] It seems as if the singularity at $z=1$ for $\alpha=0$ had moved along the real axis through the black circle to $\infty$ thereby producing an infinite number of secondary white and black lobes (or fingers) confined to a wedge shaped region with opening angle $\alpha\pi/2$ .

[646.1.1] The behaviour of $\mbox{\rm E}_{{\alpha,\beta}}(z)$ for $0<\alpha<2$ is generally dominated by the wedge $\mathbb{W}^{+}(\alpha\pi/2)$ indicated by dashed lines in Figure 4. [646.1.2] For $z\in\mathbb{W}^{+}(\alpha\pi/2)$ the Mittag-Leffler function grows to infinity as $|z|\to\infty$ . [646.1.3] Inside this wedge the function oscillates as a function of $\operatorname{Im}z$ . [646.1.4] For $z\in\mathbb{W}^{-}(\alpha\pi/2)$ the function decays to zero as $|z|\to\infty$ . [646.1.5] Along the delimiting rays, i.e. for $\arg(z)=\pm\alpha\pi/2$ , the function approaches $1/\alpha$ in an oscillatory fashion.

Figure 4: Contour plot for $\operatorname{Re}\mbox{\rm E}_{{0.2,1}}(z)$ . The dashed lines mark the wedge $\mathbb{W}^{+}(\alpha\pi/2)$ .
(The gray level coding is the same as in Figure 3)

[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 $\operatorname{Re}\mbox{\rm E}_{{\alpha,\beta}}(z)$ is symmetric with respect to the real axis.

[646.3.1] Contrary to $\mathcal{C}^{{\operatorname{Re}}}_{{0,1}}(0)$ the contour line $\mathcal{C}^{{\operatorname{Re}}}_{{0.2,1}}(0)$ consists of infinitely many pieces. [646.3.2] These pieces will be denoted as $\mathcal{C}^{{\operatorname{Re}}}_{{0.2,1}}(0;\pm k)$ 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 $\mathcal{C}^{{\operatorname{Re}}}_{{0.2,1}}(0;\pm 1)$ separates the light gray region in the left half plane from the dark gray in the right half plane. [646.3.4] The line $\mathcal{C}^{{\operatorname{Re}}}_{{0.2,1}}(0;+2)$ is the boundary of the light gray region surrounding the first white “finger” (lobe) in the upper half plane and $\mathcal{C}^{{\operatorname{Re}}}_{{0.2,1}}(0;-2)$ is its reflection on the real axis. [646.3.5] Similarly for $k=3,4,...$ . [646.3.6] Note that $\mathcal{C}^{{\operatorname{Re}}}_{{0.2,1}}(0;\pm 1)$ seem to encircle the central white lobe (“bubble”) by going to $\pm{\rm i}\infty$ parallel to the imaginary axis.

Figure 5: Truncated surface plot for $\operatorname{Re}\mbox{\rm E}_{{0.333,1}}(z)$ . Only the surface for $-1\leq\operatorname{Re}\mbox{\rm E}_{{0.333,1}}(z)\leq 1$ is shown. The contour lines $\mathcal{C}^{{\operatorname{Re}}}_{{0.333,1}}(0)$ are shown as thick solid lines. The contour lines $\mathcal{C}^{{\operatorname{Im}}}_{{0.333,1}}(0)$ are shown as thick dashed lines. Their intersections are zeros.

[646.4.1] With increasing $\alpha$ the wedge $\mathbb{W}^{+}(\alpha\pi/2)$ 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 $\mathcal{C}^{{\operatorname{Re}}}_{{\alpha,1}}(0;\pm 1)$ moves to the left. [646.4.3] This is illustrated in a threedimensional plot of $\operatorname{Re}\mbox{\rm E}_{{0.333,1}}(z)$ in Figure 5. [646.4.4] In this Figure we have indicated also the complex zeros as the intersection of $\mathcal{C}^{{\operatorname{Re}}}_{{0.333,1}}(0)$ (shown as thick solid lines) and $\mathcal{C}^{{\operatorname{Im}}}_{{0.333,1}}(0)$ (shown as thick dashed lines). [646.4.5] At $\alpha=1/2$ the contours $\mathcal{C}^{{\operatorname{Re}}}_{{0.5,1}}(0;\pm 1)$ cross the imaginary axis.

[page 647, §1]
[647.1.1] Around $\alpha\approx 0.73$ the contours $\mathcal{C}^{{\operatorname{Re}}}_{{\alpha,1}}(0;\pm 2)$ osculate the contours $\mathcal{C}^{{\operatorname{Re}}}_{{\alpha,1}}(0;\pm 1)$ . [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 $\mathcal{C}^{{\operatorname{Re}}}_{{\alpha,1}}(0;\pm 1)$ and $\mathcal{C}^{{\operatorname{Re}}}_{{\alpha,1}}(0;\pm k)$ for $k=2,3,4,...$ . [647.1.5] We estimate the value of $\alpha$ for the first osculation at $\alpha\approx 0.734375\pm 0.000015$ .

Figure 6: Contour plot for $\operatorname{Re}\mbox{\rm E}_{{0.731,1}}(z)$ and $\operatorname{Re}\mbox{\rm E}_{{0.737,1}}(z)$ , before and after the first osculation estimated to occurr at $\alpha\approx 0.734375\pm 0.000015$ . The dashed lines mark the wedge $\mathbb{W}^{+}(\alpha\pi/2)$ .
(The gray level coding is the same as in Figure 3)

[647.2.1] For $\alpha\to 1$ the dark gray fingers (where $\operatorname{Re}\mbox{\rm E}_{{\alpha,\beta}}<0$ ) extend more and more into the left half plane. [647.2.2] For $\alpha=1$ the wedge $\mathbb{W}^{+}(\pi/2)$ 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 $-\infty$ . [647.2.4] The contour lines $\mathcal{C}^{{\operatorname{Re}}}_{{1,1}}(0;\pm k)$ degenerate into

$\mathcal{C}^{{\operatorname{Re}}}_{{1,1}}(0;\pm k)=\{ z\in\mathbb{C}:\operatorname{Im}z=\pm k\pi/2\},\qquad k=1,2,3,...$

(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 $\alpha=1+\epsilon$ with $\epsilon>0$ the gray fingers are again finite. [647.2.7] This is shown in Figure 8.

Figure 7: Contour plots for $\operatorname{Re}\mbox{\rm E}_{{0.966,1}}(z)$ and $\operatorname{Re}\mbox{\rm E}_{{1,1}}(z)$ . The dashed lines mark the wedge $\mathbb{W}^{+}(\alpha\pi/2)$ .
(The gray level coding is the same as in Figure 3)

Figure 8: Contour plot for $\operatorname{Re}\mbox{\rm E}_{{1.02,1}}(z)$ . The dashed lines mark the wedge $\mathbb{W}^{+}(\alpha\pi/2)$ .
(The gray level coding is the same as in Figure 3)

[647.3.1] As $\alpha$ is increased further the fingers grow thicker and approach each other near the negative real axis. [647.3.2] For $\alpha\approx 1.42215\pm 0.00005$ the first of an infinite cascade of osculations appears. [647.3.3] This is shown in Figure 9. [647.3.4] The limit $\alpha\to 2$ 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.

Figure 9: Contour plot for $\operatorname{Re}\mbox{\rm E}_{{1.420,1}}(z)$ and $\operatorname{Re}\mbox{\rm E}_{{1.425,1}}(z)$ before and after the osculation estimated to occurr at $\alpha\approx 1.42215\pm 0.00005$ . The dashed lines mark the wedge $\mathbb{W}^{+}(\alpha\pi/2)$ .
(The gray level coding is the same as in Figure 3)

Figure 10: Contour plot for $\operatorname{Re}\mbox{\rm E}_{{1.96,1}}(z)$ . The dashed lines mark the wedge $\mathbb{W}^{+}(\alpha\pi/2)$ .
(The gray level coding is the same as in Figure 3)

Figure 11: Contour plot for $\operatorname{Re}\mbox{\rm E}_{{2,1}}(z)$ . The dashed lines mark the wedge $\mathbb{W}^{+}(\alpha\pi/2)$ .
(The gray level coding is the same as in Figure 3)

[647.4.1] For $\alpha>2$ the behaviour changes drastically. [647.4.2] Figure 12 shows the contour plot for $\alpha=3$ , $\beta=1$ . [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 $\alpha<2$ . [647.4.6] Thus the fingers are more strongly bent as they approach the negative real axis.

Figure 12: Contour plot for $\operatorname{Re}\mbox{\rm E}_{{3,1}}(z)$ . The dashed lines mark the wedge $\mathbb{W}^{+}(\alpha\pi/2)$ .
(The gray level coding is the same as in Figure 3)

[647.5.1] Now we turn to the cases $\beta\neq 1$ choosing $\alpha=1/3$ for illustration. [647.5.2] For $\beta>1$ equation (41) implies that the central lobe first grows (because $\Gamma(\beta)$ diminishes) and then shrinks as $\beta\to\infty$ for small values of $\alpha$ . [647.5.3] This is illustrated in the upper left subfigure of Figure 13. [647.5.4] For reference the case $\beta=1$ is also shown in the upper right subfigure of Figure 13.

Figure 13: Contour plot for $\operatorname{Re}\mbox{\rm E}_{{0.333,\beta}}(z)$ with $\beta=3,1,2/3,1/3$ . The dashed lines mark the wedge $\mathbb{W}^{+}(\alpha\pi/2)$ .
(The gray level coding is the same as in Figure 3)

[647.6.1] More interesting behaviour is obtained for $\beta<1$ . [647.6.2] In this case the contours $\mathcal{C}^{{\operatorname{Re}}}_{{1/3,\beta}}(0;\pm 1)$ 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 $\mathcal{C}^{{\operatorname{Re}}}_{{1/3,\beta}}(0;\pm k)$ and $\mathcal{C}^{{\operatorname{Re}}}_{{1/3,\beta}}(0;\pm(k+1))$ begins starting from $k=\infty$ . [647.6.5] One of the last of these osculations can be seen for $\beta=1/3$ on the lower right subfigure of Figure 13. [647.6.6] As $\beta$ falls below $1/3$ the contour $\mathcal{C}^{{\operatorname{Re}}}_{{1/3,\beta}}(0;+1)$ coalesces with $\mathcal{C}^{{\operatorname{Re}}}_{{1/3,\beta}}(0;-1)$ to form a new large finite central lobe. [647.6.7] This new second lobe becomes smaller and retracts towards the origin for $\beta\to 0$ . [647.6.8] This can be seen from the upper left subfigure of Figure 14 where the case $\beta=0$ is shown. [647.6.9] As $\beta$ 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 $\beta$ crosses a negative integer and is a consequence of the poles in $\Gamma(\beta)$ .

Figure 14: Contour plot for $\operatorname{Re}\mbox{\rm E}_{{1/3,\beta}}(z)$ for $\beta=0,-2/3,-1,-4/3$ . The dashed lines mark the wedge $\mathbb{W}^{+}(\alpha\pi/2)$ .
(The gray level coding is the same as in Figure 3)

Authors:	R. Hilfer und H.J. Seybold
originally published in:	Integral Transforms and Special Functions Vol. 17, 637-652 (2006)
originally submitted:	March 21st, 2005