13 Confluent Hypergeometric Functions Kummer Functions 13.8 Asymptotic Approximations for Large Parameters 13.10 Integrals

§13.9 Zeros

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Referenced by:: §13.22
Permalink:: http://dlmf.nist.gov/13.9
See also:: Annotations for Ch.13

§13.9(i) Zeros of $M\left(a,b,z\right)$
§13.9(ii) Zeros of $U\left(a,b,z\right)$

§13.9(i) Zeros of $M\left(a,b,z\right)$

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Keywords:: Kummer functions, asymptotic approximations, distribution, inequalities, number of, zeros
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)).
Referenced by:: Erratum (V1.0.5) for References
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.
See also:: Annotations for §13.9 and Ch.13

If and $b-a\neq 0,-1,-2,\dots$ , then $M\left(a,b,z\right)$ has infinitely many -zeros in $\mathbb{C}$ . When $a,b\in\mathbb{R}$ the number of real zeros is finite. Let be the number of positive zeros. Then

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

, $b\geq 0$ ,

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Symbols:: $\left\lceil\NVar{x}\right\rceil$ : ceiling of and : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(i), §13.9 and Ch.13

13.9.2

$a\geq 0$ , $b\geq 0$ ,

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Symbols:: : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(i), §13.9 and Ch.13

13.9.3

$a\geq 0$ ,

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Symbols:: : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(i), §13.9 and Ch.13

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

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Symbols:: $\left\lfloor\NVar{x}\right\rfloor$ : floor of and : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(i), §13.9 and Ch.13

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

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Symbols:: $\left\lceil\NVar{x}\right\rceil$ : ceiling of and : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(i), §13.9 and Ch.13

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

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Symbols:: $\left\lceil\NVar{x}\right\rceil$ : ceiling of , $\left\lfloor\NVar{x}\right\rfloor$ : floor of and : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(i), §13.9 and Ch.13

The number of negative real zeros is given by

13.9.7

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Defines:: : number of negative real zeros (locally)
Symbols:: : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(i), §13.9 and Ch.13

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

13.9.8 $\phi_{r}=\frac{j_{b-1,r}^{2}}{2b-4a}\left(1+\frac{2b(b-2)+j_{b-1,r}^{2}}{3(2b-% 4a)^{2}}\right)+O\left(\frac{1}{a^{5}}\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : 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
See also:: Annotations for §13.9(i), §13.9 and Ch.13

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

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

For fixed $a,b\in\mathbb{C}$ the large -zeros of $M\left(a,b,z\right)$ satisfy

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer and : complex variable
Referenced by:: §13.9(i)
Permalink:: http://dlmf.nist.gov/13.9.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(i), §13.9 and Ch.13

where 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 $M\left(a,b,z\right)$ 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 and $a\leq b<\ifrac{2a}{(1-a)}$ . The same results apply for the th partial sums of the Maclaurin series (13.2.2) of $M\left(a,b,z\right)$ .

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

For fixed and in $\mathbb{C}$ the large -zeros of $M\left(a,b,z\right)$ are given by

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, : nonnegative integer and : complex variable
Referenced by:: §13.9(i)
Permalink:: http://dlmf.nist.gov/13.9.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(i), §13.9 and Ch.13

where is a large positive integer.

For fixed and in $\mathbb{C}$ the function $M\left(a,b,z\right)$ has only a finite number of -zeros.

§13.9(ii) Zeros of $U\left(a,b,z\right)$

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Keywords:: Kummer functions, number of, zeros
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).
Referenced by:: Erratum (V1.0.5) for References
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.
See also:: Annotations for §13.9 and Ch.13

For fixed and in $\mathbb{C}$ , $U\left(a,b,z\right)$ has a finite number of -zeros in the sector $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta(<\tfrac{3}{2}\pi)$ . Let be the total number of zeros in the sector $|\operatorname{ph}z|<\pi$ , be the corresponding number of positive zeros, and , , and be nonintegers. For the case $b\leq 1$

13.9.11 $T(a,b)=\left\lfloor-a\right\rfloor+1,$

, $\Gamma\left(a\right)\Gamma\left(a-b+1\right)>0$ ,

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\left\lfloor\NVar{x}\right\rfloor$ : floor of and : number of zeros
Permalink:: http://dlmf.nist.gov/13.9.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(ii), §13.9 and Ch.13

13.9.12 $T(a,b)=\left\lfloor-a\right\rfloor,$

, $\Gamma\left(a\right)\Gamma\left(a-b+1\right)<0$ ,

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\left\lfloor\NVar{x}\right\rfloor$ : floor of and : number of zeros
Permalink:: http://dlmf.nist.gov/13.9.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(ii), §13.9 and Ch.13

13.9.13

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Symbols:: : number of zeros
Permalink:: http://dlmf.nist.gov/13.9.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(ii), §13.9 and Ch.13

and

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

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Symbols:: $\left\lceil\NVar{x}\right\rceil$ : ceiling of and : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(ii), §13.9 and Ch.13

13.9.15

$a+1\geq b$ .

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Symbols:: : number of positive zeros
Permalink:: http://dlmf.nist.gov/13.9.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §13.9(ii), §13.9 and Ch.13

For the case $b\geq 1$ we can use and .

In Wimp (1965) it is shown that if $a,b\in\mathbb{R}$ and , then $U\left(a,b,z\right)$ has no zeros in the sector $|\operatorname{ph}{z}|\leq\frac{1}{2}\pi$ .

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

For fixed and in $\mathbb{C}$ the large -zeros of $U\left(a,b,z\right)$ are given by

13.9.16 $a=-n-\frac{2}{\pi}\sqrt{zn}-\frac{2z}{\pi^{2}}+\tfrac{1}{2}b+\tfrac{1}{4}+% \frac{z^{2}\left(\frac{1}{3}-4\pi^{-2}\right)+z-(b-1)^{2}+\frac{1}{4}}{4\pi% \sqrt{zn}}+O\left(\frac{1}{n}\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, : nonnegative integer and : complex variable
Referenced by:: §13.9(ii), DLMF Update; Version 1.0.13, Erratum (V1.0.12) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16)
Permalink:: http://dlmf.nist.gov/13.9.E16
Encodings:: TeX, pMML, png
Errata (effective with 1.0.13):: In applying changes in Version 1.0.12 to this equation, an editing error was made; it has been corrected.
Reported 2016-09-12 by Adri Olde Daalhuis
Errata (effective with 1.0.12):: Originally this equation was expressed in terms of the asymptotic symbol $\sim$ . As a consequence of the use of the order symbol on the right hand side, $\sim$ was replaced by .
Reported 2016-07-11 by Rudi Weikard
See also:: Annotations for §13.9(ii), §13.9 and Ch.13

where is a large positive integer.

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

13.8 Asymptotic Approximations for Large Parameters 13.10 Integrals

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§13.9 Zeros

Contents

§13.9(i) Zeros of

§13.9(ii) Zeros of

§13.9(i) Zeros of $M\left(a,b,z\right)$

§13.9(ii) Zeros of $U\left(a,b,z\right)$