Closed form rational approximations to Tricomi psi functio

Fields, J. L.

Wimp, J.

English

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i3YERI.M REPORT 
June 1968 
MRI Project NO. 3162-p 
Contract No. NAS 9-7641 
For 
NASA Manned Spacecraft Center 
General Research Procurement Branch 
Houston, Texas 77058 
Attn : J. W. Carls on/BG731(48 ) 
425 VOLKER BOULEVARD/KANSAS CITY, MISSOURI 64110/AC 816 LO 1-0202 
https://ntrs.nasa.gov/search.jsp?R=19690005618 2020-03-12T08:26:09+00:00Z
RATIONAL APPROXIMATIONS TO T.RIC0MI'S Y F"I0N 
Je r ry  Fields 
Jet Wimp 
INTERIM REPORT 
June 1968 
MRI Project NO. 3162-P 
Contract No. NAS 9-7641 
For 
NASA Manned Spacecraft Center 
General Research Procurement Branch 
Houston, Texas 77058 
Attn : J . W. Carlson/BG731( 48 ) 
b+ 1 
425 VOLKER BOULEVARD/KANSAS CITY, MISSOURI 64110/AC 816 LO 1-0202 
PREFACE 
This report ,  wr i t ten  by Jerry Fields and Je t  Wimp, Mathematics 
Branch, Midwest Research Ins t i t u t e ,  covers work performed from 20 March 
through 19 June 1968 on Contract No. MAS 9-7641. 
Approved f o r  : 
MIDWEST RESEARCH INSTITUTE 
Sheldon L. Levy, Direeltor 
Mathematics and Physics Division 
1 J u l y  1968 
ii 
I n  t h i s  paper w e  derive closed form r a t i o n a l  approximations t o  the 
Tricomi Y function ([I-]), 
Re  a > 0 , Iarg(ei"pv,) < n/2 , -n < cp < n , 
which converge uniformly on compact subsets of the sector  
v # 0 . 
G-function ([l] ) 
larg v I < n/2 , 
A s  Tricomi's Y function can be wr i t ten  i n  terms of the Meijer 
we ac tua l ly  develop r a t i o n a l  approximations t o  t h e  G-function 
l-al, 1-CY2 
E ( v )  = Gk;;(v-'l ) 
( 3  ) 
-S - " ~ ( - s ) T ( s + q ) I ' ( s + a 2 ) v  ds , larg v \  < 3n/2 , 
2ni  
where the contour L runs from -i- t o  +i- , and separates t he  poles of 
r(-s) from those of i7(s+al)I'(s+a2), see 11.1. If a1-a2 is not an integer, 
it follows from the  residue theorem t h a t  
Our r a t i o n a l  approximations for E(v) are obtained as follows. 
If the contour L i n  (3) i s  moved k u n i t s  t o  t h e  r igh t ,  w e  obtain 
1 
r ( -S ) r (s+i )r ( s+n+al)r ( s+n+a2 ) -s v as , r( s+L+n) 
As  it can be shown that Rk(v) = O(V-~) as v-+w and larg V I  < 3n/2 , 
(5)  i s  a paraphrase of the statement 
Multiplying the f i rs t  l i ne  of ( 5 )  by a rb i t r a ry  
k = 0 t o  a fixed integer 
krk and summing from 
n , we obtain the equations 
Then we see that  
E(v )  and Fn(v,Y)/hn('Y) is  i t s  corresponding error .  The triangular form 
Jrn(v7Y) 
Jrn(v,y)/hn(Y.) 
can a l so  be wr i t ten  as 
is  a formal r a t iona l  approximation t o  
2 
I n  Fields [2], the above formulation w a s  shown t o  be equivalent 
t o  the Lanczos 7-method, see [ 3 ] ,  and the  following theorem w a s  proved. 
Theorem 1. I f  l a s g  V I  < n/2 , v # 0 , and 
then 
As the  asymptotic estimate 
was already known, see [4], the  proof of Theorem 1 reduced t o  obtaining a 
proper estimate fo r  Fn(v,v) . This w a s  effected by showing t h a t  the d i f -  
f e r e n t i a l  operator which annihilates E(v) , 
when applied t o  F,(v,Y) yields 
3 
Thus, i f  t h e  
(L3) i s  e s sen t i a l ly  a Jacobi polynomial which has a uniform algebraic rate 
of gruwth i n  n , @(no) , f o r  0 5 Y/v I; 1 . 
technique then implies 
a re  chosen as indicated i n  (S), the  right-hand s ide of 
A var ia t ion  of parameters' 
F,(v,v) =@(no)  , n - + m  , v f ixed . (14 1 
Note t h a t  i n  Theorem 1 , the  parameter h is  e s sen t i a l ly  unspecified. 
By special iz ing h and considering difference instead of d i f f e r e n t i a l  op- 
erators ,  we obtain, among other benefi ts ,  a more convenient formulation of 
the e r r o r  Fn(v,v) . 
Let  
where E- j  is  the s h i f t  operator on n , i.e., E - j  f ( n )  = f ( n - j )  , and { I  
3 
n(n+orl)(nw2 )(n+h-3 I3 , + [ A ~ + ~ B ~  E- , = 
j =1 1 J ]  (2n+A-3 )3 
Equation (16 ) concluded next page. 
4 
(n-1 ) (n+h-3 ) (2n+h -2 )2 (n+h -12~-3 ) (n+A-a2-3 ) 
( 2 n+ A -5 ) ( n+al ) ( n+a2 ) 
- 
( n+h -2 
J 
Theorem 2. If the k a re  chosen so tha t  
J 
then 
Proof. A l l  these r e su l t s  follow d i rec t ly  by computation from the 
operator equations 
5 
Corollary 2.1. If i n  Theorem 2, A-3 is  a negative integer, then 
Hence, t o  analyze the e r ror  Fn(v,v) , i n  t h i s  case, it is suf- 
f i c i e n t  t o  analyze the equation 
To do t h i s ,  we introduce some recent r e su l t s  of Wimp, [5] . Let 
, 
r ( -S )r ( -S -a1 )r ( -S -a2 ) (n+h )s 
w'ds , - 1  
(n+l)+ 
then Wimp's works shows t h a t  
* This is not qui te  the function tha t  Wimp t rea ted  i n  [SI. But a close 
inspection of t h a t  work shows t h a t  his  analysis is ac tua l ly  applicable. 
The ident i f ica t ion  process with Wimp's work i s  made by replacing h i s  
h,  n and Y by w, n+h and -h , respectively.  Also, t h e  applica- 
t i o n  of Lemma 3 i n  t h i s  reference is made eas ie r  by employing the f a c t  
2dl-% = 1+Y . 
6 
This leads t o  OUT main r e su l t ,  
Theorem 3. If larg vl < n/2 , v # 0 , then gn(veni) , gn(vewni) 
and hn(v) 
equation (21). 
as defined by (17)  form a basis  of solutions for the  difference 
Proof. It follows from (11) and (23) tha t  the  three functions 
are  l inear ly  independent as  functions of n i n  the  r igh t  half  plane. A 
d i r ec t  computation using IhF in t eg ra l  representation i n  (22) and an analog 
of (19)  shows t h a t  gn(ve-nl) s a t i s f y  (21). 
Corollary 3.1. I f  
-n, n+h 
hn(v) = I-) h = 1 or 2 , l+a ,1+01* 
and 
C'(v) , C+(V) independent of n such t h a t  
larg vi < n/2 , v # 0 , then there ex i s t  w e l l  defined analyt ic  functions 
7 
Proof. Cleazly, t he  left-hand s ide of (25) is  j u s t  the e r r o r  
From Corollary 2.1 and Theorem 3 it follows tha t  there  
CB(v) , 8 = +, -, 0, i n  
Fn(v,v)/hn(v) . 
e x i s t  functions I arg v 1 < rr/2 , such t h a t  
The functions Ce(v) can be found i n  theory by s e t t i n g  n equal t o  zero, 
one and then two in  t h e  f i r s t  l i n e  of (25) and then solving the resu l t ing  
equations. The l inear  independence of gn(vemni) , gn(ve+'ri) and hn(v) 
implies the  ana ly t ic  character of the 
From Theorem 1, we deduce t h a t  Co(v) 
Equation (26) then reduces t o  the  f irst  l i n e  of (25). 
C"(v) i n  
is iden t i ca l ly  zero i n  
la rg  V I  < v/2 , v # 0 . 
J a rg  v I < n/2 . 
Tne last  l i n e  of (25) 
follows from the estimates and the simple f a c t ,  
fieicn16 = 1 + e 
Corollary 3.2. The sequence of r a t i o n a l  approximations i n  
on compact sub- ) 
1 -a1 J 1-0?2 
Corollary 3.1 converges uniformly t o  G;;: c-11 0 
s e t s  of J arg V I  < rr/2 , v # o . 
Finally,  we r e i t e r a t e  the  main advantages of the r a t iona l  approxi- 
mations i n  Corollary 3.1. F i r s t ,  they axe e x p l i c i t ,  as opposed t o  mini-max 
r a t i o n a l  approximations which, i n  general, can only be given numerically. 
Second, they can be computed f a i r l y  easi ly ,  due t o  the f a c t  t h a t  both num- 
era tor  and denominator polynomials, $,(v,v) hn(v) , s a t i s f y  t h e  same 
t h i r d  order recursion r e l a t ion .  And last, an e x p l i c i t  form fo r  the e r ror  
i s  known. 
and 
8 
BIBLIOGRAPHY 
1. A. Erdglyi, et al., Higher Transcendental Functions, Vol. I, McGraw-Hill, 
New York, 1953. 
2. J. L. Fields, "Rational Approximations to Generalized Hypergeometric 
Functions, Math. Comp., - 19 (1965), 606-624. 
3. C. Ianczos, 
1956. 
4. J.L. Fields 
ge ome tr i c 
Applied Analysis, Prentice-Hall, Englewood CLiffs, N.J., 
and Y.L. Luke, "Asymptotic Expansions of a Class of Hyper-  
Polynomials with Respect to the Order, 11," J. Math. Anal. 
Appl., 1 (1963), 440-451. 
5. J. Wimp, "The Asymptotic Representation of a Class of G-Functions for 
Large Pmameter, '' Math. Comp., - 21 (1967), 639-646. 
9 
MIDWEST RESEARCH INSTITUTE/425 VOLKER BOULEVARD/KANSAS CITY, MISSOURI 64110 


Rational approximations to Tricomi's psi function Interim report, 20 Mar. - 19 Jun. 1968

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