§13.9 Zeros

Referenced by:: §13.22
Permalink:: http://dlmf.nist.gov/13.9

§13.9(i) Zeros of $\mathop{M\/}\nolimits\!\left(a,b,z\right)$
§13.9(ii) Zeros of $\mathop{U\/}\nolimits\!\left(a,b,z\right)$

§13.9(i) Zeros of $\mathop{M\/}\nolimits\!\left(a,b,z\right)$

Notes:: See Buchholz (1969, §17), Erdélyi et al. (1953a, §6.16), Slater (1960, Chapter 6), and Tricomi (1950a). The proof of (13.9.8) is given in Tricomi (1947). (13.9.9) follows from (13.7.2). For the paragraph following (13.9.9) see Andrews et al. (1999, §4.16). For (13.9.10) use (13.8.9) and the asymptotics of zeros of Bessel functions (§10.21(vi)).
Keywords:: Kummer functions, inequalities
Referenced by:: Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/13.9.i
Addition (effective with 1.0.5):: The reference to Segura (2008) has been added near the end of this subsection.
Reported 2012-07-30

If $a$ and $b-a\neq 0,-1,-2,\dots$ , then $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ has infinitely many $z$ -zeros in $\Complex$ . When $a,b\in\Real$ the number of real zeros is finite. Let $p(a,b)$ be the number of positive zeros. Then

13.9.1 $p(a,b)=\left\lceil-a\right\rceil,$ $a<0$ , $b\geq 0$ ,

Symbols:: $\left\lceil x\right\rceil$ : ceiling of $x$ and $p(a,b)$ : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E1
Encodings:: TeX, pMML, png

13.9.2 $p(a,b)=0,$ $a\geq 0$ , $b\geq 0$ ,

Symbols:: $p(a,b)$ : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E2
Encodings:: TeX, pMML, png

13.9.3 $p(a,b)=1,$ $a\geq 0$ , $-1<b<0$ ,

Symbols:: $p(a,b)$ : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E3
Encodings:: TeX, pMML, png

13.9.4 $p(a,b)=\left\lfloor-\tfrac{1}{2}b\right\rfloor-\left\lfloor-\tfrac{1}{2}(b+1)\right\rfloor,$ $a\geq 0$ , $b\leq-1$ .

Symbols:: $\left\lfloor x\right\rfloor$ : floor of $x$ and $p(a,b)$ : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E4
Encodings:: TeX, pMML, png

13.9.5 $p(a,b)=\left\lceil-a\right\rceil-\left\lceil-b\right\rceil,$ $\left\lceil-a\right\rceil\geq\left\lceil-b\right\rceil$ , $a<0$ , $b<0$ ,

Symbols:: $\left\lceil x\right\rceil$ : ceiling of $x$ and $p(a,b)$ : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E5
Encodings:: TeX, pMML, png

13.9.6 $p(a,b)=\left\lfloor\tfrac{1}{2}\left(\left\lceil-b\right\rceil-\left\lceil-a\right\rceil+1\right)\right\rfloor-\left\lfloor\tfrac{1}{2}\left(\left\lceil-b\right\rceil-\left\lceil-a\right\rceil\right)\right\rfloor,$ $\left\lceil-b\right\rceil>\left\lceil-a\right\rceil>0$ .

Symbols:: $\left\lceil x\right\rceil$ : ceiling of $x$ , $\left\lfloor x\right\rfloor$ : floor of $x$ and $p(a,b)$ : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E6
Encodings:: TeX, pMML, png

The number of negative real zeros $n(a,b)$ is given by

13.9.7 $n(a,b)=p(b-a,b).$

Symbols:: $p(a,b)$ : number of positive zeros and $n(a,b)$ : number of negative real zeros
Permalink:: http://dlmf.nist.gov/13.9.E7
Encodings:: TeX, pMML, png

When $a<0$ and $b>0$ let $\phi _{r}$ , $r=1,2,3,\dots$ , be the positive zeros of $\mathop{M\/}\nolimits\!\left(a,b,x\right)$ arranged in increasing order of magnitude, and let $j_{{b-1,r}}$ be the $r$ th positive zero of the Bessel function $\mathop{J_{{b-1}}\/}\nolimits\!\left(x\right)$ (§10.21(i)). Then

13.9.8 $\phi _{r}=\frac{j_{{b-1,r}}^{2}}{2b-4a}\left(1+\frac{2b(b-2)+j_{{b-1,r}}^{2}}{3(2b-4a)^{2}}\right)+\mathop{O\/}\nolimits\!\left(\frac{1}{a^{5}}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $r=1,2,3,\dots$ , $\phi _{r}$ : positive zeros and $j_{{b,r}}$ : positive zero of Bessel
Referenced by:: §13.22, §13.9(i)
Permalink:: http://dlmf.nist.gov/13.9.E8
Encodings:: TeX, pMML, png

as $a\to-\infty$ with $r$ fixed.

Inequalities for $\phi _{r}$ are given in Gatteschi (1990), and identities involving infinite series of all of the complex zeros of $\mathop{M\/}\nolimits\!\left(a,b,x\right)$ are given in Ahmed and Muldoon (1980).

For fixed $a,b\in\Complex$ the large $z$ -zeros of $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ satisfy

13.9.9 $z=\pm(2n+a)\pi i+\mathop{\ln\/}\nolimits\!\left(-\frac{\mathop{\Gamma\/}\nolimits\!\left(a\right)}{\mathop{\Gamma\/}\nolimits\!\left(b-a\right)}\left(\pm 2n\pi i\right)^{{b-2a}}\right)+\mathop{O\/}\nolimits\!\left(n^{{-1}}\mathop{\ln\/}\nolimits n\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.9(i)
Permalink:: http://dlmf.nist.gov/13.9.E9
Encodings:: TeX, pMML, png

where $n$ is a large positive integer, and the logarithm takes its principal value (§4.2(i)).

Let $P_{{\alpha}}$ denote the closure of the domain that is bounded by the parabola $y^{2}=4\alpha(x+\alpha)$ and contains the origin. Then $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ has no zeros in the regions $P_{{\ifrac{b}{a}}}$ , if $0<b\leq a$ ; $P_{{1}}$ , if $1\leq a\leq b$ ; $P_{{\alpha}}$ , where $\alpha=\ifrac{(2a-b+ab)}{(a(a+1))}$ , if $0<a<1$ and $a\leq b<\ifrac{2a}{(1-a)}$ . The same results apply for the $n$ th partial sums of the Maclaurin series (13.2.2) of $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ .

More information on the location of real zeros can be found in Zarzo et al. (1995) and Segura (2008).

For fixed $b$ and $z$ in $\Complex$ the large $a$ -zeros of $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ are given by

13.9.10 $a=-\frac{\pi^{2}}{4z}\left(n^{2}+(b-\tfrac{3}{2})n\right)-\frac{1}{16z}\left((b-\tfrac{3}{2})^{2}\pi^{2}+\tfrac{4}{3}z^{2}-8b(z-1)-4b^{2}-3\right)+\mathop{O\/}\nolimits\!\left(n^{{-1}}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.9(i)
Permalink:: http://dlmf.nist.gov/13.9.E10
Encodings:: TeX, pMML, png

where $n$ is a large positive integer.

For fixed $a$ and $z$ in $\Complex$ the function $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ has only a finite number of $b$ -zeros.

§13.9(ii) Zeros of $\mathop{U\/}\nolimits\!\left(a,b,z\right)$

Notes:: See Buchholz (1969, §17), Erdélyi et al. (1953a, §6.16), Slater (1960, Chapter 6), and Tricomi (1950a). For (13.9.16) use (13.8.10) and the asymptotics of zeros of Bessel functions (§10.21(vi)). For the final paragraph of this subsection apply Kummer’s transformation (13.2.39) to the final term in the connection relation (13.2.42) and then use the asymptotic relation (13.8.1).
Keywords:: Kummer functions
Referenced by:: Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/13.9.ii
Addition (effective with 1.0.5):: The reference to Segura (2008) has been added near the end of this subsection.
Reported 2012-07-30

For fixed $a$ and $b$ in $\Complex$ , $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ has a finite number of $z$ -zeros in the sector $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{3}{2}\pi-\delta(<\tfrac{3}{2}\pi)$ . Let $T(a,b)$ be the total number of zeros in the sector $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ , $P(a,b)$ be the corresponding number of positive zeros, and $a$ , $b$ , and $a-b+1$ be nonintegers. For the case $b\leq 1$

13.9.11 $T(a,b)=\left\lfloor-a\right\rfloor+1,$ $a<0$ , $\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left\lfloor x\right\rfloor$ : floor of $x$ and $T(a,b)$ : number of zeros
Permalink:: http://dlmf.nist.gov/13.9.E11
Encodings:: TeX, pMML, png

13.9.12 $T(a,b)=\left\lfloor-a\right\rfloor,$ $a<0$ , $\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)<0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left\lfloor x\right\rfloor$ : floor of $x$ and $T(a,b)$ : number of zeros
Permalink:: http://dlmf.nist.gov/13.9.E12
Encodings:: TeX, pMML, png

13.9.13 $T(a,b)=0,$ $a>0$ ,

Symbols:: $T(a,b)$ : number of zeros
Permalink:: http://dlmf.nist.gov/13.9.E13
Encodings:: TeX, pMML, png

and

13.9.14 $P(a,b)=\left\lceil b-a-1\right\rceil,$ $a+1<b$ ,

Symbols:: $\left\lceil x\right\rceil$ : ceiling of $x$ and $P(a,b)$ : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E14
Encodings:: TeX, pMML, png

13.9.15 $P(a,b)=0,$ $a+1\geq b$ .

Symbols:: $P(a,b)$ : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E15
Encodings:: TeX, pMML, png

For the case $b\geq 1$ we can use $T(a,b)=T(a-b+1,2-b)$ and $P(a,b)=P(a-b+1,2-b)$ .

In Wimp (1965) it is shown that if $a,b\in\Real$ and $2a-b>-1$ , then $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ has no zeros in the sector $|\mathop{\mathrm{ph}\/}\nolimits{z}|\leq\frac{1}{2}\pi$ .

Inequalities for the zeros of $\mathop{U\/}\nolimits\!\left(a,b,x\right)$ are given in Gatteschi (1990). See also Segura (2008).

For fixed $b$ and $z$ in $\Complex$ the large $a$ -zeros of $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ are given by

13.9.16 $a\sim-n-\frac{2}{\pi}\sqrt{zn}-\frac{2z}{\pi^{2}}+\tfrac{1}{2}b+\tfrac{1}{4}+\frac{z^{2}\left(\frac{1}{3}-4\pi^{{-2}}\right)+z-(b-1)^{2}+\frac{1}{4}}{4\pi\sqrt{zn}}+\mathop{O\/}\nolimits\!\left(\frac{1}{n}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\sim$ : asymptotic equality, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.9(ii)
Permalink:: http://dlmf.nist.gov/13.9.E16
Encodings:: TeX, pMML, png

where $n$ is a large positive integer.

For fixed $a$ and $z$ in $\Complex$ , $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ has two infinite strings of $b$ -zeros that are asymptotic to the imaginary axis as $|b|\to\infty$ .

§13.9 Zeros

Contents

§13.9(i) Zeros of

§13.9(ii) Zeros of

§13.9(i) Zeros of $\mathop{M\/}\nolimits\!\left(a,b,z\right)$

§13.9(ii) Zeros of $\mathop{U\/}\nolimits\!\left(a,b,z\right)$