This document discusses confluent hypergeometric functions, including: 1) Definitions of Kummer and Whittaker functions and their properties such as integral representations, connections to Bessel functions, recurrence relations, asymptotic expansions, and special cases. 2) Methods for numerically calculating zeros, turning points, and graphs of these functions. 3) References for further information. It also includes tables of values for the confluent hypergeometric function M(a,b,z) and zeros of M(u,b,z).

1. 13. Confluent Hypergeometric Functions
LUCY
JOAN
SLATEB'
Contents
Page
Mathematical Properties . . . . . . . . . . . . . . . . . . . 504
13.1. Definitions of Kummer and Whittaker Functions . . . . . 504
13.2. Integral Representations . . . . . . . . . . . . . . . 505
13.3. Connections With Bessel Functions . . . . . . . . . . . 506
13.4. Rkcurrence Relations and DifTerential Properties . . . . . 506
13.5. Asymptotic Expansions and Limiting Forms . . . . . . . 508
13.6. Special Cases . . . . . . . . . . . . . . . . . . . . 509
13.7. Zeros and Turning Values . . . . . . . . . . . . . . . 510
Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 511
13.8. Use and Extension of the Tables . . . . . . . . . . . . 511
13.9. Calculation of Zeros and Turning Points . . . . . . . . . 513
13.10. Graphing M(a. b. z) . . . . . . . . . . . . . . . . . . 513
References . . . . . . . . . . . . . . . . . . . . . . . . . . 514
Table 13.1. Confluent Hypergeometric Function M(a. b. z) . . . . . 516
Z= . l(.1) l(1) 10. a= -1 (. 1)l. b= . 1 (.l)l, 8s
Table 13.2. Zeros of M(u. b. z) . . . . . . . . . . . . . . . . . . 535
~=-1(.1)-.1, b=.I(.l)I, 7D
The tables were calculated by the author on the electronic calculator EDSACI in the
Mathematical Laboratory of Cambridge University. by kind permission of its director. Dr .
hl . V . Wilkes. The table of M(a. b. 2) was recomputed by Alfred E . Beam for uniformity
to eight significant figures.
* University Mathematical Laboratory. Cambridge. (Prepared under contract with the
N a t i O d Bureau O StaIldlUdS.)
f

2. 13. Confluent Hypergeometric Functions
Mathematical Properties
13.1. Definitions of Kummer and Whittaker U(a, b, z) is a many-valued function. Its princi-
Functions pal branch is given by -rag
<r z 5 r.
Kummer's Equation Logarithmic Solution
13.1.6
?+bz
! ( -) dw
--uw=o
13.1.1 dZ dz
It has aregular singularity at z=O and an irregular
singularity at m . (?I-l)! -
Independent solutions are +- z "M(a-n, 1-n, 2)"
(a)
I?
Kummer's Function for n=OJ 1, 2, . . ., where the last function is the
13.1.2 sum to n terms. It is to be interpreted as zero
when n=O, and ~(a)=I"(a)/I?(a).
13.1.7 U(a, 1-n, z)=z"U(a+q., l+n, z)
where
As 9 z + m
( u ) ~ = u ( u + ~ ) ( u +.~.)(u+n-1), (u)~=I,
.
13.1.8 U(a, b, z)=z-"[l+O(l~J-')]
and Analytic Continuation
13.1.3 13.1.9
U(a, b, ze*")=-
?r
sin ?rb
e-'
' M(b-a, b, Z)
(1 +a- b) r (b)
I?
-efrf(l-B) z1-b M(1-a,2--bJ Z)
r (a)r (2- b) 1
Parameters
(m, n positive integers) M(a, b, 4 where either upper or lower signs are to be taken
b#-n a#-m a convergent series for throughout.
all values of a, b and z
b#-n a=-m a polynomial of degree m 13.1.10
in2
b=-n a#-m
b=-n a=-m, a simple pole at b=-n +e-hfbRU(u, z)
b,
m>n
Alternative Notations
b=-n a=-m, undefined IFl(a;b; z) or @(a;b; z) for M(a, b, z)
m5n z-"2Fo(a,+a-b; ;- l/z) or *(a; b; z) for V(a, b, z)
1
U(a, b, z) is defined even when b+fn
As 1 2 1 + m J Complete Solution
13.1.4 13.1.11 p=AM(a, b, z)+BU(a, b, Z)
M(a, b, z)=m
r (a)e'z"-b[l+O(lzl-l)] (9z>O)
where A and B are arbitrary constants, b#-n.
Eight Solutions
and
b,
13.1.12 ~,=M(u, Z)
13.1.5
13.1.13 ys=zl-bM(l+a-b, 2-b, Z)
13.1.14 y,=ezM(b--a, b, -2)

3. CONFLUENT HYPERGEOMETRIC FUNCTIONS 505
13.1.15 y4=z1-bezM(1--a,2-b, -2) 13.1.34
13.1.17 ~ s = ~ ' - ~ U ( l + a - b ,
2-b, Z)
General Confluent Equation
13.1.18 y,=e'U(b--a, b, -2)
13.1.35
13.1.19 y,=zl-bezU(l--a, 2-4 w"+[ 2A bh'
-+2j'+--h'-77 h" ]w'
-2)
z h
WmIMkianS
h" A(A-1)
If W{m, n} =y,y~--y,,y& and
t=sgn ( J z ) = l if .fz>o,
=-1 if Y Z l O
13.1.20 Sohtions:
W{1, 2}=W{3,4}=W{l,4}=-W{2, 3) 13.1.36 Z-Ae-f(z)M(a,b, h(Z))
= (1 4 ) z - bez
13.1.21 13.1.37 Z-Ae-f")U(a, b, h(Z))
W{1, 3}=W{2,4}=W{5,
6}=W{7,8}=0 13.2. Integral Representations
13.1.22 W{1, 5)=-r(b)~-~e~/r(a) 9b>Wa>O
13.2.1
13.1.23 W{1, 7)= r(b)e"~*~-~e~/r(b--a)
13J.24
}
W{2, 5) = -r(2 -b)z- "*/r(+a- 3)
13.1.25 W{2, 7 = -r(2 -b)z- bez/r(
13.1.26 W(5, 7}=e"f(b-a'-* e
z
1
1 -a)
Kummer Transformations 13.2.2
13.1.27 M(a, b, z)=e'M(b--a, b, -2)
13.1.28
13.2.3
~'-~M(l+a-b,
2-4 z)=zl-*ezM(l-u, 2-b, -2)
13.1.29 U(a, b, 2)=z1-*U(l+a-6, 2 1 6 , Z)
13.2.4
13.1.30
Whittaker's Equation
13.1.31 %+I-,+-+ 1 K (t-r')],=,
zz
Solutions:
13.2.6
Whittaker's Functions
13.1.32 Mg,,(z)=e-+'z++,M(3+p-N, 1+2p, Z)
13.1.33
13.2.7
Wg.,(Z)=e-W+W( 3+p--K, 1 +2p, 2)
(-r<arg ZIT,~ - up=+b-i)
N=+ ,

4. 506 CONFLUENT HYPERGEOMETRIC FmycmoN8
13.3.7
13.2.8 r (a)~ ( a ,Z)
b,
= eAzJAm e-z,(,-A)a-'(t+B)b-a-l~t
(A=l-B)
Smlr integrals for ME,&) and W#,,,(z) can
iia
be deduced with the help of ..13.1.32'and 13.1.33.
Barnes-typeContour Integrals
13.2.9
for larg (-z)l<)r, a, b#O, -1, -2, . . . . The
contour must separate the poles of I'(-s) from
those of r(a+s); c is finite.
=e& 3 C"z"(-~2)'"-"")~b--l+n(2~(--az))
m
n-
13.2.10
where
r(a)r(i+a-b)z"u(a, b, Z)
1 c+im Co=l, c,= -bh, c,=-)(2h-l)a+ib(b+l)hZ,
-- r(-8)r(~+~)r(i+a-b+s)z-*ds
-2d L-t m (n+l)Cs+l= (1 -2h)n--bhJC,
[
3r +[(1 -2h)a -h(h -1) (b +n- 1)]Cn-1
for larg z1<2, a#O, -1, -2, . . ., b - a f l , 2,
-h(h- l)aC,-2 (h real)
3, . . . . The contour must separate the poles of
r(--s) from those of r(a+s) and r(l+a-b+s).
13.3. Connections With b e e l Functions where
(see chapters 9 and 10) c,=1, C,(a, b)=2a/b,
Beace1 Functiom M LIrniti- caseo Cn+da, b)=2aC,(a+l, b+l)/b-Cs-,(a, b)
If b and z are fixed, 13.4. Recurrence Relations and Merentid
Properties
13.3.1 h { M ( ab, z/a)/r(b)}
~ =2*-'1)-1(2m
a+- 13.4.1
13.3.2 lim {M(a,b,-z/a)/r(b)} =z4-'Jb-1(21/z) (b-a)M(a-1, b, z)+(2a-b+z)M(a, b, 2)
a+-
13.3.3 -aM(a+l, b, z)=O
b {r(l+a-b) U(a, b, z/a)}=2z4-'&-1(2a 13.4.2
a+-
b(b-l)M(a, b-1, z)+b(l-b-z)M(a, b, 2)
13.3.4 +z(b-a)M(a, b+l, z)=O
lim{l"(l+a-b)U(a, b, -z/a)} 13.4.3
a+-
= --xieTibz4-4bH$1(2~ (./z>O) (l+a-b)M(a, b, 2)-aM(u+l, b, 2)
13.3.5 -,,.ie-rib&+bHC21
- b-1(2&) O(
)z
<J +(b-l)M(a, b-1, z)=O
in Seriem
E S ~ I ~ O M 13.4.4
13.3.6 bM(a, b, 2)-bM(a-1, b, 2)-zM(a, b + l , z)=O
M(a, b, z)=e**r(b-~-))(tz)"-~++
13.4.5
b(a+z)M(a, b, z)+z(a-b)Ai(a, b+l, 2)
--abM(a+l, b, z)=O

5. {
CONFLUENT HYPERGEOMETRIC FUNCTIONS 507
13.4.6 13.4.19
(a+z)U(a, b, z)-zU(a, b+l, z)
(a-l+z)M(a, b, z)+(b-a)M(a-l, b, 2)
+(1-b)M(a, b-1, z)=O +a@-a-l)U(a+l, b, 2)=0
13.4.20
13.4.7
(a+z-l)U(a, b, 2)-U(a-1, b, z)
b(l-b+z)M(a, b, z)+b(b-l)M(a-1, b-1, Pi +(l+a-b)U(a, b-1, z)=O
-azM(a+l, b + l , z)=O
13.4.21 U'(U, b, z)=-aU(a+l, b+l, Z)
13.4.22
13.4.9
d"
- M(a, b,
dz" Z) } (a)"M(a+n, b+n, 2)
13.4.23
13.4.10 aM(a+l, b, z)=aM(~, z)+zM'(a, b,
b, Z) a(l+a-b)U(a+l, b, z)=aU(a, b, 2)
+zU'@, b, 2)
13.4.11
13.4.24
(b-a)M(a-1, b, z)=(b-a-z)M(a, b, 2) (l+a-b)U(a, b-1, z)=(l-b)U(a, b, 2)
+zM'(a, b, 4 -zU'(a, b, 2)
13.4.12
13.4.25 U(a, b + l J z)=U(U,b, z)-U'(U, b, Z)
(b-a)M(a, b + l , z)=bM(a, b, z)-bM'(a, b, 2)
13.4.26
13.4.13 U(a-1, b, z)=(a-b+z)U(a, b, z)-zU'(a, b, 2)
(b-l)M(a, 6-1, z)=(b-l)M(a, b, 2)
13.4.27
+zM'(a, bJ z
, U(u-1, b-1, z)=(l--b+z)U(a, b, 2)
13.4.14 -zU'(a, b, 2)
(b-l)M(u-l, 6-1, z)=(b-1-z)M(a, b, 2)
+zM'(aJ bJ z,
13.4.15
U(a-1, b, z)+(b-2a-z)U(a, b, 2)
+a(l+a-b)U(a+l, b, z)=O
13.4.16
(b--a-l)U(fZ, b-1, z)+(l-b-z)U(a, b, 2)
+zU(a, b+l, z)=O
13.4.11
U(a, b, 2)-aU(a+l, b, 2)-U(a, b-1, z)=O
13.4.18
(b-a) U(a, b, 2) + U(a- 1, b, 2)
-zU(a, b + l , z)=O

6. 508 C0"T HYPEBGEOMETIUC FUNCI'IONS
13.5.11 (b=O)
13.5.12
a
~LS 4-m for b bounded, z real.
where u ie defined in 13.5.13.
aa a+-- for b bounded, x rad.
For large real a, b x
,
If cdah' 6 = ~ / ( 2 b - 4 ~ ) 80 that ~>2b-U>l,

7. 13.5.19
13.5.20
{ Z-2b-4~
U(a, b, z)=e+=+"-+T(+) T-+&-*
CONFLUENT HYPERGEOMETRIC F"CM0NB
If z= (2b-&)[l+t/(b--2a)~], so that
M(a, b, z)=e+=(b-2a)'-Or(b)[Ai(t) cos (UT)
+Bi(t) sin (UT) +O(I4b-a I-*)]
i--tr(~)(bz--2az)-13f~-f+O(l~--al-i)}
If cos*f?=z/(2b-4~) that 2b--4a>z>O,
so I
13.5.21
13.5.22
U(U,b, ~)=exp
13.6. SpeCi.1 Casea
[(b-24
{
M(a, b, z)= r(b) exp (b-24 COS* e}
[(b-2~) COSe]'-'[~($b-u) sin m]-+
[sin (ad+sin (+-a) (2e-sin 2e) +ir)
C O S ~ B ] [ ( ~ - ~ U ) COS
[(3b-U) sin 2e)-*{sin[($&a)
(20- sin 26)+ t TI + O(l 3b--al-') 1
509
el1-*
Relation Function
13.6.1 BeeSel
13.6.2 &See1
13.6.3 Modified Bessel
13.6.4 Spherical Besael
13.6.5 Spherical Besael
13.6.6 Spherical Besael
13.6.7 Kelvin
13.6.8 Coulomb Wave
13.6.9
13.6.10 Incomplete Gamma
13.6.11 Poisson-Charlier
13.6.12 e* Exponential
13.6.13 Trigonometric
13.6.14 Hyperbolic
13.6.15
Weber
13.6.16
or
Parabolic Cylinder
13.6.17 Hermite
13.6.18 Hermite
13.6.19 Error Integral
13.6.20 *
Toronto
*See page 11.

8. 510 CONFLUENT HYPEROEOMETBIC F"C"I0NS
13.6. Spedrl CuebGntinued
Relation Function
a b
13.6.B V+t 2v+ 1 22 Modified Bessel
13.6.22 V+t 2v+ 1 -2ir Hankel
13.6.23 V+t 2v+ 1 2it Hankel
13.6.24 n+l 2n+2 2s Spherical Bessel
13.6.25 9 + 42'1' Airy
13.6.26 n+t 2n+1 6 Kelvin
13.637 -n a+1 2 Lsguem
13.6.28 1--a 1--a 2 Incomplete Gamma
13.6.29 1 1 -2 Exponential Integral
13.6.30 1 1 2 Exponential Integrgl
13.6.31 1 1 --In z Logarithmic Integral
13.6.32 tm-n l f m 2 Cunningham
13.6.33 -t V 0 22 Bateman
13.6.34 1 1 iz Sine and Cosine Integral
13.6.35 1 1 -iz Sine and Cosine Integral
{
13.6.36 -t V t t* Weber
or
13.6.37 4-1. t 42' Parabolic Cylinder
13.6.38 t-tn t a9 Hermite
13.6.39 t t 39 Error Integral
13.7. &roe and Turning Values For the derivative,
If jD-l,, the r'th positive zero OfJ&l(z), then
is 13.7.4
a first approximation Xo to the r'th positive zero
}
of M(a, b, z) is
13.7.1 XO=~:-~,,
1/(2b-4a)+0(1/(3b-u)2)
13.7.2 If X is the first approximation to a turning value
L
of M(u, b, z), that is, bo a zero of M'(u, b, z) then
a better approxiniation is
A closer approximation is given by
13.7.3 Xl=XO-M(a, 6,Xo)/M'(u,b, Xo)

9. CONFLUENT HYPERGEOMETRIC FU"I[ONS 51 1
The self-adjoint equation 13.1.1 can ala0 be is an increasing or decreasing function of z, that is,
written they form an increasing sequence for M(a, b, z)
if a>O, z -
<$
b or if a<O, z>b-$, and a decreas-
13.7.6 ing sequence if a>O and z>b-3 or if a O and
<
zb$
< -.
The Sonine-Polya Theorem
The maxima and minima of Iwl form an in-
I The turning values of Il lie near the curves
w
creasing or decreasing sequence according as
-e-'e-& I
Numerica1 Methods
13.8. Use and Extension of the Tables In this way 13.4.1-13.4.7 can be used together
Calculation of M(a, b, x) with 13.1.27 to extend Table 13.1 to the range
Kummer's Transformation -10<a<10, -10 j b <lo, -10 <z<10.
E x m p l e 1. Compute M(.3, .2, -.I) to 7s. This extension of ten units in any direction is
Using 13.1.27 and Tables4.4 and 13.1 we have possible with the loss of about 1s. Al the re-
l
a = & b=.2 so that currence relations are stable except i) if a<O, O
b
<
M(.3, .2, -.1) =e-.'M(-.l, .2, .l) and lal>lbl, z>O, or ii) b a
<, b<O, Ib--al>lbl,
z<O, when the oscillations may become large,
=.85784 90. especially if II also is large.
z
Thus 13.127 can be used to extend Table 13.1 to Neither interpolation nor the use of recurrence
negative values of z Kummer's transformation
. relations should be attempted in the strips
should also be used when a and b are large and b=-nf.1 where the function is very large nu-
nearly equal, for z large or small. merically. In particular M(a, b, z) cannot be
Example2. Compute M(17, 16, 1) to 7s. evaluated in the neighborhood of the points
Here a=17, b=16, and a=-m, b=-n, m j n , as near these points
M(17, 16, l)=elM(-l, 16, -1) small changes in a, b or z can produce very large
changes in the numerical value of M(a, b, z).
=2.71828 18X1.06250 00
Example 4. At the point (- 1, -1, z),M(u, b, z)
=2.88817 44. is undefined.
R e c u r r m a Relations 2
When a=-1, M(-1, b, z)=l-afor all 2.
Example 3. Compute M(--1.3, 1.2, .l) to 7 .
s
Using 13.4.1 and Table 13.1 we have a=-.3, Hence lim M(-1, b,z)=l +z. ButM(b,b,z)=e)
b+-1
b=.2 so that for all z when a=b. .Hence lim M(b, 6, z)=&.
,
b+-1
M(-1.3, .2, .1)=2[.7 M(-.3, .2, .1) -.3 M(.7, .2, .l)] In the first case b+- 1 along the line a=-1, and
=.35821 23. in the second case b+-1 along the line a=b.
By 13.4.5 when a=-1.3 and b= .2, Derivatives
M(-1.3,1.2, .1)=[.26 M(--3, .2, .l) Example 5. To evaluate M'(-.7, -.6, .5) to
-.24 M(--1.3, .2, .1)]/.15 7s. By 13.4.8, when a= -.7 and b= -.6, we have
=A9241 08. -.7
M'(-.7, -.6, .5)=- M(.3, .4, .5)
-.6
Similarly when a=-.3 and b= .2
=1.724128.
M(-.3, 1.2, .1)=.97459 52.
Asymptotic Formulas
Check, by 13.4.6,
For ~ 2 1 0 a and b small, M(a, b, z) should be
,
M(-1.3, 1.2, .1)=[.2 M(-.3, .2, .l) evaluated by 13.5.1 using converging factors
4-1.2 M(-.3, 1.2, .1)]/1.5 13.5.3 and 13.5.4 to improve the accuracy if
=A9241 08. necessary.

10. 512 CONFLUENT HYPERGEOMETRIC FUNCTIONS
Example 6. Calculate M(.9, .l, 10) to 7S, Hence
using 13.5.1. U(.1, .2, 1) =5.344799(.371765-.194486)
= .94752.
Similarly
U(-.9, .2, 1)=.91272.
Hence by 13.4.15
=-.198(.869) +1237253(.99190 285) U(l.1, 2, l)=[U(.l, .2, l)-U(-.9, .2, 1)]/.09
+ O(1) = .38664.
= 1227235.23- .17 +O(1)
= 1227235+0(1) Example 10. To compute U'(-.9, - . 8 , 1) to
5s. By 13.4.21
Check, from Table 13.1, M(.9, .l, 10)=1227235.
To evaluate M(a, b, z) with a large, z small and b U'(-.9, -.8, 1)=.9U(.1, .2, 1)
small or large 13.5.13-14 should be used. = (.9)(.94752)
Example7. Compute M(-52.5, .l, 1) to 3s, =.85276.
using 13.5.14. Asymptotic Formulae
M(-52.5, .l, 1) = r(.l)e-'(.05+52.5).25-.M Example 11. To compute U(1, .l, 100) to 5s.
.5642 COS [(.2-4( -52.5)) . I - .05r+ .254 By 13.5.2
11 +0((.05+52.5)-a6)]= -16.34+0(.2) 1 19 1929
U(1, .l, 100)=i&j{l-:+:
By direct application of a recurrence relation, 100 100100
M(-52.5, .l, 1) has been calculated as -16.447.
To evaluate M(a, b, z) with z, a and/or b large,
13.5.17,19 or 21 should be tried.
Example8. Compute M(-52.5, .1, 1) using =.01{1-.019+.000551-.000021
13.5.21 to 3s, COS e2
=4
.
' +0(10-9) 1,
=.00981 53.
M(-52.5, .l, 1)
Example 12. To evaluate V(.l, .2, .01). For
-r(.l)e*oJ.l 1105.1 COS 8J1-.*.5641
- coa2e
z small, 13.5.612 should be used.
52.55-1 sin 28-1[& (-52.5~)
r (1 -.2)
+sin (52.55(2e-~in 2e)+tr) U(.l, .2, .Ol)=
r (1.1 -.2) +O((.01)1- -7
+ O((52.55)-')]= -16.47-t O(.02)
=-+O( (.01)-7
A full range of asymptotic formulas to cover all U.9)
possible cases is not yet known. =1.09 to 3S, by 13.5.10.
Calculation of U(a, b, x)
To evaluate U(u, b, z) with a large, z small and
For - 1 0 5 ~ 5 1 0 , - 1 0 5 ~ 5 1 0 , -105b510 b small or large 13.5.15 or 16 should be used.
this is possible by 13.1.3, using Table 13.1 and the To evaluate V(a, b, z) with z, a and/or b large
recurrence relations 13.4.15-20. 13.5.18, 20 or 22 should be tried. In all these
Example 9. Compute U(l.1, .2, 1) to 5s. cases the size of the remainder term is the guide to
Using Tables 13.1, 4.12 and 6.1 and 13.1.3, we the number of significant figures obtainable.
have
Calculation of the Whittaker Functiona
U(.1, .2, 1)=
Example 13. Compute M.o.-.4(l) W.o,-.4(1)
and
to 5s. By formulas 13.1.32 and 13.1.33 and
Tables 13.1, 4.4
But M(.9, 1.8, 1)=.8[M(.9, . 8 ,l)-M(-.l, .8, l)] -44.0, -.,(l) =e-,'M(.l, .2, 1)=1.10622,
= 1.72329, using 13.4.4. W.o.-.,(1) =e-.(U( .1, .2, 1) = .57469.

11. Thus the values of M..,(z) and W d z ) can x;=x;
[l- M‘(-3,.6,Xi)
always be found if the values of M(a, b, z) and -3M(-3, .6, Xi)1
U(a, b, z) are known. =Xi [I-M(-2,1.6, Xi)/.6M(-3, .6,Xi)]
13.9. Calculation of Zeros and Turning Points =.9715)<1.0163=.9873 to 4s.
Ex-Ple 14. a m p u b the smallest POsitive This process can be repeated to give as many
zero of M(-4, . 6 , ~ > .This is outside the range of significant figures as are required.
Table 13.2. Using 13.7.2 we have, as a first
approximation
If we repeat this calculation, we find that
FIGURE13.1.
X2=X1+.00002 99=.17852 99 to 7s.
Figure 13.1 shows the curves on which M(a, 6, z)
Calculation of Marima and Minima =O in the a, b plane when z=1. The function is
positive in the unshaded areas, and negative in the
Examp1e15* Compute the va1ue Of z at which
shaded areas. The number in each square gives
M(-1.8’-*2’z) a turningvalue’ Using13*4*8
has
the number of real positive zeros of &&, b, z) as a
and Table 13.2, we find that M’(-1.8, -.2,2)
function of z in that square. The vertical
=9M(-.8, . 8 , z)=O when x=.94291 59.
boundaries to the left are to be included in each
Also M”(-1.8, -.2, z)=9M’(-.8, .8, 2)’
square.
-9M(.2, 1.8, z) and M(.2, 1.8, .94291 59)>0.
13.10. Graphing M(a, b, x)
Hence M(--1.8, -.2, z) has a maximum in z when
~=.94291 59. Example 17. Sketch M(-4.5, 1, z). Firstly,
Example 16. Compute the smallest positive from Figure 13.1 we see that the function has
value of x for which M(-3, .6, z) has a turning five real positive zeros. From 13.5.1, we find
value, Xi. This is outside the range of Table 13.2. that M+- m , M’+- m as x++ m and that
Using 13.4.8 we have M++m, M’++m as z+--. By 13.7.2 we
have 6s first approximationsto the zeros, .3,1.5,3.7,
M’(-3, .6, ~)=-3M(-2, 1.6, ~)/.6. 6.9, 10.6, and by 13.7.2 and 13.4.8 we find as first
approximations to the turning values .9, 2.8, 5.8,
By 13.7.2 for M(-2, 1.6, z), 9.9. From 13.7.7, we see that these must lie near
X =(1.0!k)2/(11.2)= .9715.
o the curvea
y = f eN(54-t (1 -dl l)%-+.
Thisisafirst approximationto XiforM(-3, .6,z).
Using 13.7.5 and 13.4.8 we find a second approxi- From these facts we can form a rough graph of
mation the behavior of the function, Figure 13.2.

12. 514 CONFLUENT HYPERGEOMETRIC F"Cl'I0NS
FIQUF~E
13.2. M(-4.5, 1, 2.
)
(From F. Gb2d?'ri, R~m$d;~y&~*&~o~l*
Edblonl.
FIQUBE
13.4. M(a, .5, 2.
)
(Ffom E. J8hnke cmd F. Emde Table8 of hmctlons Dover Publlcatknu,
Inc, New York, fi.Y., 1945, with p m b l o n . )
References
Tcxts
[13.11 H. Buchholz, Die konfluente hypergeometrische
Funktion (Springer-Verlag, Berlin, Germany,
1953). On Whittaker functions, with a large
bibliography.
(13.21 A. Erdelyi et al., Higher transcendental functions,
vol. 1, ch. 6 (McGraw-Hill Book Co., Inc., N e w
York, N.Y., 1953). On Kummer functions.
[13.3] H. Jeffreys and B. 5. Jeffreys, Methods of mathe-
matical physics, ch. 23 (Cambridge Univ. P, -
Cambridge; England, 1950). On Kummer
functions.
[13.4] J. C. P. Miller, Note on the general solutions of the
confluenthypergeometric equation, Math. Tablea
Aids Comp. 9,97-99 (1957).
FIQWE13.3. M(o, 1, z). 113.61 L. J. Slater, On the evaluation of the confluent
hypergeometric function, Proc. Cambridge
(prom E. Jahnke and F Emde Tables of function& Dover Publlatkxu.
ha., New York, &.Y, lM6, with pemmbsbm.) Philoe. Soc. 49, 612-622 (1953).

13. CONFLUENT HYPERQEOMETBIC FUNCNONS 515
[13.6] L. J. Slater, The evaluation of the basic confluent (13.121 J. R. Airey and H. A. Webb, The practical impor-
hypergeometric function, Proc. Cambridge tance of the confluent hypergeometric function,
Philos. Soc. 50, 404-413 (1954). Phil. Mag. 36, 129-141 (1918). M(a, b, z),
[13.7] L. J. Slater, The real mros of the confluent hyper- ~=-3(.5)4, b=1(1)7, z=1(1)6(2)10, 45.
geometric function, Proc. Cambridge Philos. (13.131 E. Jahnke and F. Emde, Tables of functions, ch. 10,
Soc. 52, 626-635 (1956). 4th ed. (Dover Publications, Inc., New York,
[13.8] C. A. Swanson and A. Erdhlyi, Asymptotic forms N.Y., 1945). Graphs of M(a, b, z) based on the
of confluent hypergeometric functions, Memoir tables of [13.11].
25, Amer. Math. S o c .(1957). [13.14] P. Nath, Confluent hypergeometric functions,
[13.9] F. G. Tricomi, Funeioni ipergeometriche confluenti Sankhya J. Indian Statist. Soc. 11, 153-166
(Edizioni Cremonese, Rome, Italy, 1954). On (1951). M(u, b, z), a=1(1)40, b=3,2=.02(.02)
Kummer functions. .1(.1)1(1)10(10)50, 100, 200, 6D.
[13.10] E. T. Whittaker and G. N. Watson, A course of [13.15] 8. Rushton and E. D. Lang, Tables of the confluent
modern analysis, ch. 16, 4tb ed. (Cembridge hypergeometric function, Sankhye J, Indian Statist.
Univ. Press, Cambridge, England, 1952). On So c. 369-411 (1954). M(a, b, Z) , a=.5(.5)40,
13,
Whittaker functions. b= .5(.5)3.5, Z= .02 (.02).1 (.1) 1 (1) 10(10)50, 100,
200, 7s.
T d h [13.16] L. J. Slater, Confluent hypergeometric functions
(Cambridge Univ. Preas, Cambridge, England,
[13.11] J. R. Airey, The confluent hypergeometricfunction, 1960). M(u, b, z), ~ = - l ( . l ) l , b=.l(.l)l,
British Association Reports, Oxford, 276-294 ~=.l(.l)lO, 8s; M(u, b, l), ~=-11(.2)2,
(1926), and Lee&, 220-244 (1927). M(a, b, z), b= -4(.2) 1, 85; and smallest positive values of
~=-4(.5)4, a=*, 1, 3, 2, 3, 4, ~=.1(.1)2(.2)3 z for which Mfa, b, z)=O, a=-4(.1)-.l,
(.5)8, 5D. b=.1(.1)2.5, 8s.