13 . Confluent Hypergeometric Functions LUCY JOAN SLATEB' Contents Mathematical Properties ................... 13.1. Definitions of Kummer and Whittaker Functions ..... 13.2. Integral Representations ............... 13.3. Connections With Bessel Functions ........... 13.4. Rkcurrence Relations and DifTerential Properties ..... 13.5. Asymptotic Expansions and Limiting Forms ....... 13.6. Special Cases .................... 13.7. Zeros and Turning Values ............... Page 504 504 505 506 506 508 509 510 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 Table 13.2. Zeros of M(u. b. z) .................. 535 Z= .l (.1) l(1) 10. a= - 1 (. 1)l. b= . 1 (.l)l, 8s ~=-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 . (Preparedunder contract with the NatiOd Bureau Of StaIldlUdS.)
Transcript
13 . Confluent Hypergeometric Functions LUCY JOAN SLATEB'
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 NatiOd Bureau Of StaIldlUdS.)
If XL is the first approximation to a turning value of M(u, b, z), that is, bo a zero of M'(u, b, z) then a better approxiniation is
CONFLUENT HYPERGEOMETRIC FU"I[ONS 51 1
The self-adjoint equation 13.1.1 can ala0 be written
13.7.6
I The Sonine-Polya Theorem
The maxima and minima of Iwl form an in- creasing or decreasing sequence according as
I - e - ' e - & Numerica
13.8. Use and Extension of the Tables Calculation of M(a, b, x) Kummer's Transformation
Compute M(.3, .2, -.I) to 7s. Using 13.1.27 and Tables4.4 and 13.1 we have a=& b=.2 so that
M( .3, .2, - .1) =e-.'M( - .l, .2, .l)
Thus 13.127 can be used to extend Table 13.1 to negative values of z. Kummer's transformation should also be used when a and b are large and nearly equal, for z large or small.
Example2. Compute M(17, 16, 1) to 7s. Here a=17, b=16, and
By 13.4.5 when a= - 1.3 and b= .2, M(-1.3,1.2, .1)=[.26 M(--3, .2, .l)
-.24 M(--1.3, .2, .1)]/.15 =A9241 08.
Similarly when a= - .3 and b= .2
M(-.3, 1.2, .1)=.97459 52.
Check, by 13.4.6, M(-1.3, 1.2, .1)=[.2 M(-.3, .2, .l)
4-1.2 M(-.3, 1.2, .1)]/1.5 =A9241 08.
is an increasing or decreasing function of z, that is, they form an increasing sequence for M(a, b, z) if a>O, z<b-$ or if a<O, z>b-$, and a decreas- ing sequence if a>O and z>b-3 or if a<O and
The turning values of Iwl lie near the curves z<b-$.
1 Methods In this way 13.4.1-13.4.7 can be used together
with 13.1.27 to extend Table 13.1 to the range
-10<a<10, -10 j b <lo, -10 <z<10.
This extension of ten units in any direction is possible with the loss of about 1s. All the re- currence relations are stable except i) if a<O, b<O and lal>lbl, z>O, or ii) b<a, b<O, Ib--al>lbl, z<O, when the oscillations may become large, especially if IzI also is large.
Neither interpolation nor the use of recurrence relations should be attempted in the strips b=-nf.1 where the function is very large nu- merically. In particular M(a, b, z) cannot be evaluated in the neighborhood of the points a= - m, b=-n, m j n , as near these points small changes in a, b or z can produce very large changes in the numerical value of M(a, b, z).
Example 4. At the point (- 1, - 1, z), M(u, b, z) is undefined. When a=-1, M(-1, b, z)=l-afor all 2.
Hence lim M(-1, b,z)=l +z. ButM(b,b,z)=e)
for all z, when a=b. .Hence lim M(b, 6, z)=&.
In the first case b+- 1 along the line a= - 1, and in the second case b+-1 along the line a=b.
2
b+-1
b+-1
Derivatives
Example 5. To evaluate M'(-.7, -.6, .5) to 7s. By 13.4.8, when a= - .7 and b= -.6, we have
For ~ 2 1 0 , a and b small, M(a, b, z) should be evaluated by 13.5.1 using converging factors 13.5.3 and 13.5.4 to improve the accuracy if necessary.
512 CONFLUENT HYPERGEOMETRIC FUNCTIONS
Example 6. Calculate M(.9, .l, 10) to 7S, using 13.5.1.
=-.198(.869) +1237253(.99190 285)
= 1227235.23- .17 + O(1) = 1227235+0(1)
+ O(1)
Check, from Table 13.1, M(.9, .l, 10)=1227235. To evaluate M(a, b, z) with a large, z small and b small or large 13.5.13-14 should be used.
Example7. Compute M(-52.5, .l, 1) to 3s, using 13.5.14.
M( -52.5, .l, 1) = r( .l)e-'(.05 + 52.5).25-.M .5642 COS [(.2-4( -52.5)) . I - .05r+ .254
11 +0((.05+52.5)-a6)]= -16.34+0(.2)
By direct application of a recurrence relation, 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.
Compute M(-52.5, .1, 1) using Example8. 13.5.21 to 3s, COS e=4'2.
M(-52.5, .l, 1) - -r(.l)e*oJ.l coa2e 1105.1 COS 8J1-.*.5641
52.55-1 sin 28-1[& (-52.5~) +sin (52.55(2e-~in 2e)+tr)
+ O((52.55)-')]= - 16.47-t O(.02)
A full range of asymptotic formulas to cover all possible cases is not yet known.
Calculation of U(a, b, x)
For - 105~510 , - 1 05~510 , -105b510 this is possible by 13.1.3, using Table 13.1 and the recurrence relations 13.4.15-20.
Example 9. Compute U(l.1, .2, 1) to 5s. Using Tables 13.1, 4.12 and 6.1 and 13.1.3, we have
U(.1, .2, 1)=
But M(.9, 1.8, 1)=.8[M(.9, . 8 , l)-M(-.l, .8, l)]
Example 11. To compute U(1, .l, 100) to 5s. By 13.5.2
1 1 9 1 9 2 9 100 100100 U(1, .l, 100)=i&j{l-:+:
=.01{ 1-.019+.000551-.000021
=.00981 53. +0(10-9) 1,
Example 12. To evaluate V(.l, .2, .01). For z small, 13.5.612 should be used.
+O( (.01)1- -7 r (1 - .2) r (1.1 -.2) U(.l, .2, .Ol)=
=-+O( (.01) -7 U.9)
=1.09 to 3S, by 13.5.10.
To evaluate U(u, b, z) with a large, z small and b small or large 13.5.15 or 16 should be used.
To evaluate V(a, b, z) with z, a and/or b large 13.5.18, 20 or 22 should be tried. In all these cases the size of the remainder term is the guide to the number of significant figures obtainable.
Calculation of the Whittaker Functiona
Example 13. Compute M.o.-.4(l) and W.o, -.4(1) to 5s. By formulas 13.1.32 and 13.1.33 and Tables 13.1, 4.4
Thus the values of M..,(z) and Wdz) can always be found if the values of M(a, b, z) and U(a, b, z) are known.
13.9. Calculation of Zeros and Turning Points
a m p u b the smallest POsitive
Using 13.7.2 we have, as a first
Ex-Ple 14. zero of M(-4, . 6 , ~ > . This is outside the range of Table 13.2. approxima tion
1 x;=x; [l- M‘(-3,.6,Xi) -3M(-3, .6, Xi)
=Xi [I-M(-2,1.6, Xi)/.6M(-3, .6,Xi)]
=.9715)<1.0163=.9873 to 4s.
This process can be repeated to give as many significant figures as are required.
If we repeat this calculation, we find that
X2=X1+.00002 99=.17852 99 to 7s.
Calculation of Marima and Minima
Examp1e 15* Compute the va1ue Of z at which M(-1.8’ -*2’z) has a turningvalue’ Using13*4*8 and Table 13.2, we find that M’(-1.8, -.2,2) =9M(-.8, . 8 , z)=O when x=.94291 59.
-9M(.2, 1.8, z) and M(.2, 1.8, .94291 59)>0. Hence M(--1.8, -.2, z) has a maximum in z when ~=.94291 59.
Compute the smallest positive value of x for which M(-3, .6, z) has a turning value, Xi. This is outside the range of Table 13.2. Using 13.4.8 we have
Also M”(-1.8, -.2, z)=9M’(-.8, .8, 2)’
Example 16.
M’(-3, .6, ~)=-3M(-2, 1.6, ~)/.6.
By 13.7.2 for M(-2, 1.6, z),
Xo= (1.0!k)2/(11.2) = .9715.
Thisisafirst approximation to XiforM(-3, .6,z). Using 13.7.5 and 13.4.8 we find a second approxi- mation
FIGURE 13.1.
Figure 13.1 shows the curves on which M(a, 6, z) =O in the a, b plane when z=1. The function is positive in the unshaded areas, and negative in the shaded areas. The number in each square gives the number of real positive zeros of &&, b, z) as a function of z in that square. The vertical boundaries to the left are to be included in each square.
13.10. Graphing M(a, b, x)
Example 17. Sketch M(-4.5, 1, z). Firstly, from Figure 13.1 we see that the function has five real positive zeros. From 13.5.1, we find that M+- m , M’+- m as x++ m and that M++m, M’++m as z+--. By 13.7.2 we have 6s first approximations to the zeros, .3,1.5,3.7, 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, 9.9. From 13.7.7, we see that these must lie near the curvea
y = f eN(54-t (1 -dl l)%-+.
From these facts we can form a rough graph of the behavior of the function, Figure 13.2.
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).
Inc, New York, fi.Y., 1945, with p m b l o n . ) (Ffom E. J8hnke cmd F. Emde Table8 of hmctlons Dover Publlcatknu,
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., New 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 confluent hypergeometric equation, Math. Tablea Aids Comp. 9,97-99 (1957).
113.61 L. J. Slater, On the evaluation of the confluent hypergeometric function, Proc. Cambridge Philoe. Soc. 49, 612-622 (1953).
FIQWE 13.3. M(o, 1, z).
ha., New York, &.Y, lM6, with pemmbsbm.) (prom E. Jahnke and F Emde Tables of function& Dover Publlatkxu.
CONFLUENT HYPERQEOMETBIC FUNCNONS 515
[13.6] L. J. Slater, The evaluation of the basic confluent hypergeometric function, Proc. Cambridge Philos. Soc. 50, 404-413 (1954).
[13.7] L. J. Slater, The real mros of the confluent hyper- geometric function, Proc. Cambridge Philos. Soc. 52, 626-635 (1956).
[13.8] C. A. Swanson and A. Erdhlyi, Asymptotic forms of confluent hypergeometric functions, Memoir 25, Amer. Math. S o c. (1957).
[13.9] F. G. Tricomi, Funeioni ipergeometriche confluenti (Edizioni Cremonese, Rome, Italy, 1954). On Kummer functions.
[13.10] E. T. Whittaker and G. N. Watson, A course of modern analysis, ch. 16, 4tb ed. (Cembridge Univ. Press, Cambridge, England, 1952). On Whittaker functions.
T d h
[13.11] J. R. Airey, The confluent hypergeometric function, British Association Reports, Oxford, 276-294 (1926), and Lee&, 220-244 (1927). M(a, b, z),
(13.121 J. R. Airey and H. A. Webb, The practical impor- tance of the confluent hypergeometric function, Phil. Mag. 36, 129-141 (1918). M(a, b, z),
(13.131 E. Jahnke and F. Emde, Tables of functions, ch. 10, 4th ed. (Dover Publications, Inc., New York, N.Y., 1945). Graphs of M(a, b, z) based on the tables of [13.11].
[13.14] P. Nath, Confluent hypergeometric functions, Sankhya J. Indian Statist. Soc. 11, 153-166 (1951). M(u, b, z), a=1(1)40, b=3,2=.02(.02) .1(.1)1(1)10(10)50, 100, 200, 6D.
[13.15] 8. Rushton and E. D. Lang, Tables of the confluent hypergeometric function, Sankhye J, Indian Statist. Soc. 13, 369-411 (1954). M(a, b, Z) , a=.5(.5)40, b= .5( .5) 3.5, Z= .02 (.02). 1 (. 1) 1 (1) 10( 10) 50, 100 ,
[13.16] L. J. Slater, Confluent hypergeometric functions (Cambridge Univ. Preas, Cambridge, England,
~=-3(.5)4, b=1(1)7, z=1(1)6(2)10, 45.
200, 7s.
1960). M(u, b, z), ~= - l ( . l ) l , b=.l(.l)l, ~=.l(.l)lO, 8s; M(u, b, l), ~=-11(.2)2, b= -4(.2) 1, 85; and smallest positive values of z for which Mfa, b, z)=O, a=-4(.1)-.l, b=.1(.1)2.5, 8s.
![arXiv:1101.3688v1 [math.CA] 19 Jan 2011 · arXiv:1101.3688v1 [math.CA] 19 Jan 2011 TransformationsandinvariantsfordihedralGauss hypergeometricfunctions RaimundasVidunas KobeUniversity](https://static.documents.pub/doc/80x56/5ec163748ca60d135d691dae/arxiv11013688v1-mathca-19-jan-2011-arxiv11013688v1-mathca-19-jan-2011.jpg)
