MATHEMATICS OF COMPUTATION
VOLUME 44. NUMBER 170
APRIL 19X5. PACiES 443-45«
The Generalized Integro-Exponential Function
By M. S. Milgram
Abstract. The generalized integro-exponential function is defined in terms of
integral (incomplete gamma function) and its derivatives with respect to
pendium of analytic results is given in one section. Rational minimax
sufficient to permit the computation of the first six first-order functions
another section.
the exponential
order. A comapproximations
are reported in
1. Introduction. The generalized integro-exponential function plays an important
role in the theory of transport processes and fluid flow, yet no unified summary of
its properties exists in the literature. First introduced by Van de Hülst [12] who gave
a power series and numerical tabulation for some special cases, the function was
abstracted by Chandrasekhar [8] and recently reviewed by Van de Hülst [13]. In the
interim, tabulated values were presented by Stankiewicz [26] and Gussmann [11]; the
latter as well as Abu-Shumays [2] and Kaplan [16] independently summarized some
analytic properties. Recent work by this author [20]-[22] has reasserted the significant role played by this function in transport theory, and, by invoking the theory of
Meijer's G-functions [18], many general properties of this function have been
unveiled which were previously unknown. These were summarized in an Appendix
to [20] and in a Table in a conference proceedings [22].
Since these summaries appeared, the problem of accurately evaluating this function numerically has arisen. For small values of the real independent variable x,
power series evaluations work reasonably well with coefficients obtainable from the
formulas given in [20] and [22]. For intermediate values of x, power series break
down due to cancellation of significant digits when evaluating differences between
large numbers, and for large values of x, asymptotic series do not do justice to the
inherent accuracy available with modern computers.
The purpose of this paper is then twofold:
(1) Reiterate, derive and present a unified summary of the analytic properties of
these functions only some of which can be found elsewhere;
(2) Proffer some approximations on which the numerical evaluation of a large
number of these functions can be based.
2. Analysis. Let s and z be continuous (complex) variables and n and j represent
nonnegative integers. Define the "generalized integro-exponential function" Ef(z)
by
(2-1)
_
Ei(z) = tf^Es(z),
J'
oV
Received December 1, 1981; revised April 30, 1982, October 25, 1982, November 28, 1983, February
17, 1984, and March 16, 1984.
1980 MathematicsSubjectClassification.Primary 33A70,41A20,85A25.
«'1985 American
Mathematical
Society
0025-5718/85 $1.00 + $.25 per page
443
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
M. S. MILGRAM
444
where Es(z)
is the usual* exponential
integral generalized
to n = s, or in terms of
incomplete gamma functions
Es(z)
= z°-xT(l
-s,z).
We have the integral representation
(2.2)
£°(z) = Es(z) = fX r5'exp(-z0
Ji
dt;
for the case s = n this reduces to the exponential integral about which a considerable
literature revolves [1],[3],[4],[6],[9],[14],[19],[23],[25],[27],[29].Apply Eq. (2.1) to
(2.2), interchange the order of integration and differentiation, and obtain the integral
representation
(2.3)
£/(z) = —^-
f
(log,) V'exP(-zO dt
1(1 + 1) Ji
which, when s = n, reduces to Gussmann's [11] definition En (z) = E]n(z), or that
of van de Hülst [12] E[j)(z) = E{~x(z). Equation (2.3) may be integrated by parts
(integrate t~s and differentiate the remaining factors) to obtain the recursion
formula
(2.4)
Ei(z) = (zEU(z)-EJ-\z))/(l-s),
s*l,j>0,
defining E~\z) = exp(-z) for the casey = 0.
To obtain a unified theory of this function and its properties, start with the known
identification of exp(-z) in terms of Meijer's G-function [18, No. 6.2.1(1) and 6.4(1)]
(2.5)
exp(-z) = G¿;1(z|¿.) = ¿7/
T(-t)z'dt,
substitute into Eq. (2.2), and use a known [18, No. 5.6.4(6)] integration formula to
obtain
(2.6a)
(2.6b)
H^ä,
«'>-<«('|&-1¡)-K,J[.
= exp(-z)zs~\p(s,s;
z),
where L0 is the negatively directed contour enclosing both the nonnegative r-axis,
and the pole at t = s - 1. The third term in the equality (2.6a) is the contour integral
representation [18, No. 5.2(1)] of the G-function according to prescription. In Eq.
(2.6b), ^ is a confluent hypergeometric function of the second kind. Operating on
Eq. (2.6a) as in Eq. (2.1), we obtain the contour integral representation
(2.7a)
(2.7b)
£/(z) = ¿t/
Z77/
r'-'>''
J L-n
i
i0
(s-1
^ + 2-°2|z
== G/:2-
-t)J
«
+l
;s,...,s
0,s - l,...,s
- 1;
* "Exponential
integral" (of order s) refers to the function Es(z). Some authors [13], [1] use this
nomenclature to mean En(z), while others [18] mean £",(z), or Ei(z) = -E¡(-z) or both.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
THE GENERALIZED INTEGRO-EXPONENTIAL FUNCTION
445
The result (2.7b) is obtained by identifying the contour integral (2.7a) according to
prescription [18, No. 5.2(1)]; Eq. (2.7a) provides a means of generalizing the function
to noninteger values oï j. Set y = -1 in Eq. (2.7a), and with reference to Eq. (2.5)
obtain
(2.8)
E;\z)
= exp(-z),
from which it is easy to see how Eq. (2.4) reduces to a well-known result [1, No.
5.1.14] when y = 0 and s = n.
The power series follows immediately from Eq. (2.7a) by evaluating the residues of
the simple poles of T(-t) and the multi-poles of order; + 1 when j # w ory + 2 if
s = n. The results are
(2-9)
£/(z)=E
/To (s - 1 - l)J+1l\
+ *" (,~1}E (-l)'(-;W-'*r(l
J-
i=q
Ml
- sf\
s±n,
and
,/
eHz) = y ' —^(2.10)
nK '
/£oi
(n-i-iy+1n
where T(l - s)U) means the /th derivative of T(l - s) with respect to s, and
(2.11)
'
*,„'■"
Urn ^fr(/-n
r—-l
dt'\
+ 2)r(H-Q
T(l+t)
Analytic expressions for ^ln (0 < / < 8) have been obtained through the use of
computer algebra [28]; they are given in Table 1. General, but complicated expressions can be found in [2, Eq. (4.12)]. The result (2.10) is also given in [11] and [2, Eq.
(4.7)]. Similar coefficients may be obtained for the functions E±n(z); however an
alternative method for the evaluation of the latter set of functions will be given
shortly.
It is of some interest to obtain other integral representations. From Eq. (2.7b) we
recognize the G-function as an Euler transform [18, No. 5.6.4(6)]:
(2.12)
Ej(z)= •'i r°Er\zt)dt,
;>0,
from which the name derives. The casey = 1, s = 1 has been obtained elsewhere [17]
in the form of a power series; (2.12) may also be found in [2]. An alternate
expression for Eq. (2.12) is the repeated integral form:
4W*Jf «k-'-Jf
dy]+xy-:xcxPyzUyi
From (2.2), (2.3) and (2.12), we find
(2.14a)
£>(Z)«_JT
(\ogt)J-\-°Es(zt)dt,
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
y>0,
446
M. S. MILGRAM
from which it may be shown that
/•OO
/
E's(zt)rs\ogktdt
Ji
(2.14b)
r(*rTTV1
+ l) Ir
1(1)
rk+u
Ek
+l(zt)rs\og'-ltdt,
l>0,k>0.
J\
Finally, using (2.14a) and (2.14b) we find the remarkable representation
(2.14c) Ef(z) =
1
fX EMt-tog'-'-Udt
V 0< /<y-
1 (J — t) Ji
1
by a process of induction, employing (2.19). This generalizes both (2.12) and (2.14a).
Starting from Frullani's integral
o
du(exp(-v)
- exp(-tv))/v
and Eq. (2.3), the order of integration may, with care, be interchanged, giving the
Laplace transform
(2.15)
E)(z) = fX dvcxp(-v)[Es(z)
- cxp(v)Es(z
+ v)]/v.
After applying Eq. (2.1) to Eq. (2.15), a more general form is found:
(2.16)
Er\z)
= —^—
r
(y + 1) Jo
dvexp(-v)[Ef(z)
- cxp(v)E¡(z + v)]/v,
j>0.
A number of G-function representations may also be derived. From Eq. (2.2) and
known integration formula [18, No. 5.6.2(18)], we have
Es(z)
(2-17)
= exp(-z)G12i
z
0;
|0,i-
1;,
/T(s).
Starting with Eq. (2.7a), substitute t '.= 2t and use the duplication
formula for the
gamma function to find [18, No. 5.2(1)]
(2.18)
Ef(z)
2-y-i
I ,2
;(s+
Qj+i.0
l)/2,...,(s+l)/2
0,\,(s-l)/2,...,(s-l)/2;V
From Eq. (2.3) the differentiation rules are easy to uncover
9*
-Ef(z) = (-!)* Ef_k(z),
(2.19a)
(2.19b)
9z
^EHz)={-'rTim+J
ds"'hAz)
r(y + i)
+ l)E^(z)
s
{h
and from Eqs. (2.19) and (2.4) we discover that Ej(z) satisfies
(2.20a)
(s - l)ysEf(z) - z^Ef(z)+(j
+ l)Ef(z) = 0
as well as [18, No. 5.8(1)]
(2.20b)
[l+l){zl-s
+ l)l +\Js(z) = 0
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
THE GENERALIZED INTEGRO-EXPONENTIAL FUNCTION
447
A short table of integrals is included as Appendix A.
In applications [27] we need to evaluate EJ„(x), n > 0, x = Re(z) > 0. Integrating by parts in Eq. (2.3) leads to the result
E¿(z) = E{-\z)/z,
y>0.
Repeated application of Eq. (2.4) gives
Similarly, repeated application of Eq. (2.21) leads to the simpler form
(2.22)
Ei„(z) =
r(« + i)
z"+1
"-'
z1
expt-z;
exp(-z)
E JiH* + E t&Efiz)
/= 0 ' '
OT-l
where the (precomputed) constants |/„ are given analytically by
(2.23)
É/-"
= & ^o'''io(vr7T/)lv1
ti-j..-^P;
&.-1,
+ /+>-i)'''(^T7+ir
É/..-0 if«<y+ /.
The derivation of Eq. (2.22) comes by proving its truth for the cases y = 1 and y = 2
and proceeding by induction, using the identities
n-l
(2.24a)
(2.24b)
«.„- E (j-h-)iS'.
i/,,T(r-l-7)£fill
From Eq. (2.22) all the E¿„(z) can be obtained for n > 0 if the functions Exm(z) are
known for 0 < m ^y - 1. Approximations
for Exm(z) are given in Section 3. A
lengthy form for the evaluation of Ej(x) for n > 1 in terms of E{(x) is given by
Gussmann [11, Eq. (A.18) to (A.21')]- Note that Eq. (2.22) is numerically stable since
all its terms are positive.
For large values of real z, the first few terms in the asymptotic series of E¿„(x) are
given by the first finite sum in Eq. (2.22). From Eq. (2.7b) and the theory [18,
Section 5.10] of G-functions, the asymptotic series is theoretically known. The first
terms are
(2.25)
EJ(z)~exp(-z)z^+»[l-(j+l)(j
+ 2s)/(2z)+
■■■].
A more general result can be obtained by integrating Eq. (2.12) repeatedly by parts:
(2-26)
Er(2)^i^Iil^ELk+Áz).
z k=o
zk
r(«)
The claimed asymptotic formula is then (Re(z) -» + oo)
(2.27)
EJ(z) ~ exp(-z)z--1
£ ^
t-o z
^
\+ J) fl-,,,+,+/-„
H")
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
448
M. S. MILGRAM
where P is an integer. This can be proven by establishing its truth for the case y = 0
and y = 1, using in Eq. (2.26) the well-known [1, Eq. 5.1.51] result
r
t \
I
En + k + x(z) ~ cxp(-z)z
\ -i V ('I)™ r(* + k + m + 1)
E
m=o
—i-r,
, ,
z
, .v
T(n + k + l)
,
and proceeding by induction, employing the identity
'
(2-28)
1
2rf t
i rrçii+*,n+/+/
= *n-i,n+/+/
in Eq. (2.27).
Finally, from Eq. (2.9) we have the special value
(2.29)
£/(0)=(^y)7+1,
Re(i)>l.
3. Rational Minimax Approximations to E{(x). With the aid of the computer code
REMES2 [15], it proved possible to obtain rational minimax approximations to
E{(x) for x real, 1 <y < 6, and hence EJ_n(x) with 1 <y < 7, « > 0 because of Eq.
(2.22). The region 0 < x was conveniently divided into three ranges for the purpose
of both evaluating the master functions and obtaining efficient fits.
3a. Small x. In this range, we evaluate E{(x) according to the power series (2.10)
in the form
(3.1)
E{(x) = Pk(x)/Q,(x)
+ PXj(logx),
where Pk(x)/Q¡(x) is the (k, I) minimax fit to the infinite series plus the constant
(/ =y + 1) term of the second sum in Eq. (2.10). PXj()ogx) is a polynomial in
log(eY;c) of order y obtainable from Table 1 by means of computer algebra. Although
no proof has been found, for 1 < y < 7, this polynomial PXJ(\og x) can be written
7+ 1
(3.2)
PXJ= E UjjJ,
i=i
where
(3.3)
v = log(eYx)
and
(3.4)
Ujj =-uJ_XJ_x/l,
/>1,
with the constants uJ0 given in Table 2, and y being Euler's constant. The upper end
of this range (X,j) was chosen to coincide with a point smaller than the first positive
zero of E{(x) — PXj()ogx). All master function calculations were done in 120 bit
double-precision arithmetic (-28
s.d.) and the approximation chosen was the
lowest (A: + /) entry in the Walsh array with more than 14 s.d. accuracy. The
coefficients of the (k, I) minimax fits are given in Table 3.
3b. Intermediate x. In this range, Xu < x < 10, the master functions were obtained from the power series (2.10), with an exponential weighting such that
(3.5)
E&x) = cxp(-x)Pk(x)/Q,(x).
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
THE GENERALIZED INTEGRO-EXPONENTIAL FUNCTION
449
Table 1
Analytic expressions for ^
CPSI<1>=GRMIN
*
in Eq. (2.10)
CPSI<£>=GRMIN*<-PSIN>
CPSI (3>=+GRMIN*(PSIN»*£-PSINl+£.
»ZETP2)
CPSI <4)=+GRMI!\l*<3. »PSIN*PSIN1~6.
*PSIN*ZETfl£-PSIN**3-PSIN£)
CPSI(5>=
1 +GRMIN«(4.*PSIN*PSIN2-£.*PSIN»»£*PSIN1+1£.♦
1 -12.*PSINl»ZETP£+3.
*PSIN1**£-PSIN3+1£.
PSIN»*£*ZETR£+PSII\I**4
*ZETfl£**£+l£. *ZETR4)
1
1
1
1
CPSI(6> =
+GPMIN*<60. *PSIN*PSIN1*ZETP£-15.
*PSIN»PSINl**£+5.
*PSIN*PSIN3-60.
*PSIN*ZETA£*-*£-60.
»PSIN*ZETP4-i0.
*PSIN**£*PSIN£+10.
*PSIN**3»PSIN
1-20. *PSIN*»3*ZETR£-P5IN**5+10.
*PSINl*PSIN£-£0.
*PSIN£*ZETfl2-PSIN
4)
1
1
1
1
1
1
1
CPSI(7)=
+GRMIN*(-60.«PSIN*PSIN1»PSIN£+1£0.*PSIN»PSIN£*ZETfl£+6.
*PSIN*PSIN
4-180. *PSIN**£*PSINl*ZETfl£+45.*PSIN**£*PSIN1**£-15.
*P5IN**2*PSIN
3+180.*PSIN*#£*ZETR£**2+180.*PSIN**£*ZETP4+20.*PSIN**3»PSIN£-15.
*PSIN*#4*PSINl+30.
*PSIN**4#ZETfl£+PSIN*»6+15.
*PSIN1*PSIN3-180.
*PS
IN1*ZETR£**£-180.
*PSI!\;:.*ZETP4+90.
»PSINl**£*ZETߣ-15.
*PSIN1**3+10
.*PSIN£**£-30.
*PSIN3*ZErfl£-PSIN5+3&0.
*ZETP£*ZETR4+1£0.
*ZETR£»*3+
£40. »ZETA6)
CPSI(8)=
1 +GPMIN*(-105. #PSIN*PSIM1*PSIN3+1268.
#PSIN*PSINl*ZETfl£*«2+l£6®.
1 SIN*PSINl»ZETP4-638.
*P5IN#PSIN1**2*ZET«2+1®5.
*PSIN*PSINl**3-70.
*P
*
1
1
1
1
1
1
1
1
PSIN*PSIN£**£+£10.
*PSIN*PSIN3*ZETfl£+7.*PSIN*PSIN5-25£0.»PSINa-ZET
fl£*ZETP4-B40. *PSIN*ZETP.£**3-1680.*PSIN»ZETR6+£10.
*PSIN**£*PSIN1*
PSIN2-4£0. ♦ PSIN**£*PSIN£#ZETR£-21.*PSIN**£*PSIN4+4£i2.
*PSIN**3*PS
IN1*ZETR£-105. »PSIN»*3»PSINl**£+35.*PSIN**3*PSII\!3-4£0.
*PSIN*»3»Z
ETR£**£-4£0.
*PSIN**3*ZETR4-35.*PSIN»*4*PSIN£+£1.*PSIN**5*PSIN1-4
£.*PSIN«»5*ZETP£-PSIN**7+4£0.*PSIN1*PSIN£*ZETP£+£1.*PSIN1*PSII\4105. *PSINl**£»PSIN£+35.
*PSIN£*PSIN3-4£0.*PSIN£*ZETfi£**£-4£0.*PSI
N£*ZETR4-4£. *PSIN4*ZE^A£-PSINS)
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
CPSI(9)=
+GRMIN*(-3360.
*PSIN*PS:M*PSIN£*ZETR£-168.
*PSIN*PSINl*PSIN4+840.
*PSIN*PSINl**£*PSIN£-£30.*PSIN*PSIN£*PSIN3+3360.
»PSIN*PSIN£»ZETO
£»*£+3360. »PSIN*PSIN2*ZETfl4+336.*PSIN*PSIN4*ZETR2+8.
♦PSIN*PSIN6+
420.*PSIN«»£*PSINl*PSI.\3-5040.*PSIN**2*PSINl*ZETR2**£-5040.
*PSIN
*«£*PSINl*ZETA4+25£0.
»P3IN»*£»PSINl**£»ZÊTR£-4£0.*PSIN**2*PSIN1*
*3+£80. *PSIN**£»PSIN£**£-S40.*PSIN**2*PSIN3*ZETR2-28.
*PSIN**2*PS
IN5+10080.»PSIN**£»ZET«£*ZETP4+3360.*PSIN**£*ZETfl£**3+67£0.
*PSIN
**£*ZETR6-560.*PSIN**3*PSIN1*PSIN£+11£0.
♦PSIN»»3*PSIN£*ZETfl£+56.
*PSIN**3*PSIN4-840.
*PSIN**4*PSIN1*ZETP£+£10.
*PSIN»*4*PSINl**£-70
.*PSIN*»4*PSIN3+840.
*PSIN»-*4*ZETP2»*2+840.
♦ PSIN-**4*ZETA4+5E,. »PSI
N**5*PSIN£-£B. »PSIN**6*PSINl+56.*PSIN**6*ZETfl£+PSIN«*8-£80.
*PSIN
l*PSIN£*»£+840.
*PSINl»PSIN3*ZETfi£+£8.
*PSIN1»PSIN5Î-10080.
*PSIN1*Z
ETR£*ZETR4-3360. *PSINl»ZETR£**3-£720.
»PSIN^ZETRÊ-E^.
*PSIN1**£»
PSIN3+£5£0. *PSINl**£*ZETfl£**£+£5£0.*PSINl**£*ZETR4-840.
*PSIN1**3
«ZETR2+105.*PSINl**4+56.*PSIN£*PSIN4+560.*PSIN2**£*ZETfl£-B40.
»PS
1 IN3*ZETP£**£-B40. *PSIN3*ZETR4+35.»PSIN3**£-56.*PSIN5*ZETR£-PSIN7
1 +13440.*ZETR£*ZETR6+10080.*ZETP£**£*2ETR4+16B0.♦ZETR£**4+5040.*Z
1 ETR4**£+10080.«ZETR8)
Note:
CPSI(Ul)
=^i
n
ZETAt = Ç(&)
PSIN = ip(n)
GAMIN= l./T(n)
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
PSINÎ. = * (n)
450
M. S. MILGRAM
Table 2
Coefficientsof the polynomials PXJ(log x)
coefficient u,
"lM
(numerical)
(analytic)
1
1
0
0
.822467033424113
.400685634386531
.608806818962515
.536936276078223
.5651020417 78833
.5585578221 73266
.5612749829 66818
-?(3)/3
774/160
-f(3)7r2/36 _ r(5)/5
f(3)2/18 + 61n-6/120960
-f(3)774/480 - f(5)7r2/60 - ?(7)/7
f{3)27T2/216+ f(3)J(5)/15 + 1261w8/29030400
also PXj = T.JktlxUjkvk where v = log(evx) and u]■k = -Uj_x k_x/k
example: Px3(v) = f(3)t;/3 + tt2î;2/24 + t;4/24
Table 3.1
E11(X)-P(X)/Q(X)*(X-X11)+P11(L0G(X)
IN THE RANGE 1.000.LE.X.LE.10.000
X11-.91446 48659 7277
IN THE RANGE 0.000.LE.X.LE.
1.000
RATIONALAPPROXIMATION
IS R(3,4)
WITH PRECISION- 14.46 DIGITS
P00
P01
P02
P03
Q00
Q01
Q02
Q03
Q04
(
(
(
(
6)
5)
4)
2)
6)
.376341
-.635070
-.538768
-.584198
.418437
6)
5)
3)
.121793
U
57664
75949
20580
45557
62176
FIT TO E11(X)-EXP(-X)*P(X)/Q(X)
06520
96412
92452
93210
08333
37377 23431
.132630 73906 04789
.551375 17596 10943
.100000 00000 00000
E11(X)=EXP(-X)/X**2*P(Z)/Q(Z)
; Z-l/X
IN THE RANGE 0.000.LE.Z.LE.
.100
RATIONALAPPROXIMATION
IS R(6,8)
WITH PRECISION- 14.09 DIGITS
POO
POl
P02
P03
P04
P05
1)
0)
3)
3)
3)
2)
P06
0)
Q00
Q01
Q02
Q03
Q04
Q05
Q06
Q07
Q08
0)
2)
2)
3)
3)
3)
3)
2)
RATIONALAPPROXIMATION
IS R(4,6)
WITH PRECISION- 15.13 DIGITS
POO
POl
P02
P03
P04
Q00
Q01
Q02
Q03
Q04
Q05
Q06
-2)
0)
1)
1)
1)
-2)
0)
1)
D
2)
2)
D
.368896
.111317
.111573
.430645
.522913
.368896
.122384
.144230
.747160
.168869
.135627
.100000
16541
82274
42563
72608
13537
16541
70770
98012
29601
92689
18800
00000
46809
12989
49138
32509
89389
46811
37284
73127
50458
15484
19350
00000
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
1)
-.338130
.891028
.211811
.283328
.120102
.192587
.999999
-.400917
-.106209
.419023
.569128
.964803
.616101
.175879
.222587
.100000
76971
00665
97726
58929
26074
99072
20518
24613
44376
93606
96511
06423
85034 28858
37819
25463
64057
54287
08155
47030
22239
85540
00000
68016
66475
58286
46715
39357
82591
84149
75955
00000
THE GENERALIZED INTEGRO-EXPONENTIAL FUNCTION
451
Table 3.2
FIT TO E21(X)=P(X)/Q(X)
+P12(L0G(X)
IN THE RANGE 0.000.LE.X.LE.
.400
RATIONALAPPROXIMATIONIS R(3,4)
WITH PRECISION- 16.20 DIGITS
POO (
POl (
P02
P03
Q00
Q01
Q02
Q03
Q04
(
(
(
(
(
(
(
6)
6)
5)
4)
6)
6)
4)
3)
1)
.216257
-.493516
-.784750
-.381379
-.539718
-.115308
-.773816
-.122836
.100000
42293
16951
37376
38950
43355
01225
80672
12906
00000
79812
20742
89728
85943
22324
47753
69915
79186
00000
FIT TO E21(X)=EXP(-X)*P(X)/Q(X)
IN THE RANGE
.400.LE.X.LE.
2.000
RATIONALAPPROXIMATION
IS R(5,8)
WITH PRECISION- 14.90 DIGITS
POO
POl
P02
P03
P04
P05
Q00
Q01
Q02
0.03
Q04
Q05
Q06
Q07
Q08
0)
2)
2)
2)
2)
0)
-1)
1)
2)
3)
3)
3)
3)
2)
1)
.534274
.172848
.602765
.582331
.147552
.999989
.170899
.295254
.651380
.315340
.573546
.444897
.147788
.207544
.100000
07496
04128
67841
73035
63525
73572
18403
33350
89668
67127
38415
80147
79716
71393
00000
24268
51346
77444
43360
88969
67234
89456
91985
07230
43816
89567
56000
87537
35389
00000
FIT TO E21(X)=EXP(-X)*P(X)/Q(X)
E21(X)=EXP(-X)/X**3*P(Z)/Q(Z)
; Z-l/X
IN THE RANGE 2.000.LE.X.LE.10.000
IN THE RANGE 0.000.LE.Z.LE.
.100
RATIONALAPPROXIMATIONIS R(5,8)
WITH PRECISION- 15.55 DIGITS
RATIONALAPPROXIMATION
IS R(5,5)
WITH PRECISION- 13.58 DIGITS
POO
POl
P02
P03
P04
P05
QOO
Q01
Q02
Q03
Q04
Q05
Q06
Q07
Q08
3)
3)
3)
3)
2)
0)
2)
4)
4)
.340433
.876064
.676598
.203399
4)
.778642
.547543
.194264
.352535
.306893
4)
4)
3)
2)
n
.246893
.999999
.206456
.120509
.511338
.100000
44179
53575
41845
00769
21393
81273
40742
38588
37482
56924
10587
79236
91968
01010
00000
16653
85969
71783
19242
49231
60712
55514
74781
96754
73543
36117
68901
35532
78642
00000
POO
POl
P02
P03
P04
P05
QOO
Q01
Q02
Q03
Q04
Q05
-3)
-2)
-1)
0)
-1)
-2)
-3)
-1)
0)
0)
1)
1)
.325912
.976173
.894008
.249087
.139742
-.401189
.325912
.117172
.148297
.802097
.174325
.100000
28045
27349
30387
11994
26502
07964
28045
06417
13910
98704
30019
00000
69083
42300
59118
57106
81342
43870
69169
64891
05923
24226
86266
00000
In the cases y = 5 and 6, multiple-precision (48 digit) arithmetic [7] was employed
within the master functions to overcome large losses in accuracy near the upper part
of the range, and again, the lowest (k + I) approximation with more than 14 digit
accuracy was chosen for inclusion in Table 3. In some cases, it was beneficial to split
this range into two to obtain more efficient approximations.
3c. Large x. It proved fruitless to approximate E((x) with the asymptotic formula
(2.27) in the range 10 < x. Accordingly, (k, I) minimax fits Rj(l/x) were procured
such that
(3.6)
E{(x) = exp^x-^R^l/x),
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
452
M. S. MILGRAM
Table 3.3
FIT TO E31(X)-P(X)/Q(X)+P13(LOG(X))
FIT TO E31(X)-EXP(-X)*P(X)/Q(X)
IN THE RANGE 0.000.LE.X.LE.
IN THE RANGE
.600
RATIONALAPPROXIMATION
IS R(3,4)
WITH PRECISION- 15.65 DIGITS
POO
POl
P02 (
P03 (
QOO
Q01
Q02
Q03
Q04
6)
7)
(
(
(
(
6)
5)
7)
6)
5)
3)
(
1)
-.839003
.119755
.242909
.108775
-.137811
-.296572
-.174055
-.157386
.100000
.600.LE.X-.LE.10.000
RATIONALAPPROXIMATION
IS R(5,9)
WITH PRECISION- 13.37 DIGITS
47848
64264
68230
31120
82510
03168
92800 54023
94906 03826
00000 00000
82605
65275
62547
12253
17423
26101
POO
POl
P02
P03
P04
P05
QOO
Q01
Q02
Q03
Q04
Q05
Q06
Q07
Q08
Q09
E31(X)-EXP(-X)/X**4*P(Z)/Q(Z)
; Z-l/X
IN THE RANGE 0.000.LE.Z.LE.
.100
2)
3)
3)
3)
2)
0)
0)
2)
3)
4)
4)
4)
4)
3)
2)
1)
.157208
.128368
.270646
.148490
.228734
.999998
.320385
.675468
.861412
.372041
.677596
.552376
.213346
.392231
.328733
.100000
97422
95938
30105
63549
95971
89216
15630
89806
12489
91499
67313
35994
91178
03684
75166
00000
57519
74056
87470
05473
24294
94849
95068
76635
99020
30951
99825
49519
79909
45133
04005
00000
RATIONALAPPROXIMATIONIS R(3,8)
WITH PRECISION- 14.31 DIGITS
POO
POl
P02
P03
QOO
Q01
Q02
Q03
Q04
Q04
Q05
Q06
Q07
Q08
2)
1)
0)
1)
-2)
-1)
1)
2)
2)
2)
2)
2)
0)
-.174019
-.709402
-.905606
-.359745
.174019
.883421
-.164111
-.137785
-.514428
-.514428
-.685703
-.162601
957694
100000
20356
14529
10531
25082
20356
34886
11311
23506
97426
97426
73818
49416
76113
00000
94571
05107
27851
90533
94563
03280
14874
45778
13443
13443
80260
93719
15230
00000
where 0 < 1/x < 0.1. To evaluate the master functions, Eq. (2.12) was transformed
into the integral form
(3.7)
E((x) = exp(-x)x-J-lfX
dtexp(-t)(l
+ t/x)'J~lRjÁ
t + X
where R x was a 19 digit rational minimax fit to xJexp(x)E{~l(x)
according to
Eq. (3.6) obtained especially for that purpose. The integration was a 400-point
Gauss-Laguerre calculation in double-precision arithmetic [5]. To begin the bootstrap,
high-precision fits to Ex(x) due to Cody and Thacher [9] in this range were
employed. The fit in the Walsh table with smallest (k + I) and greatest accuracy
over 14 s.d. was chosen, and is reported in Table 3.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
THE GENERALIZED INTEGRO-EXPONENTIAL FUNCTION
453
Table 3.4
FIT TO E41(X)-P(X)/Q(X)+P14(LOG(X))
FIT TO EA1(X)-EXP(-X)*P(X)/Q(X)
IN THE RANGE 0.000.LE.X.LE.
IN THE RANGE
.500
RATIONALAPPROXIMATION
IS R(3,3)
WITH PRECISION- 15.13 DIGITS
POO
POl
P02
P03
QOO
Q01
Q02
Q03
(
(
(
(
(
(
(
(
6)
7)
6)
3)
7)
6)
4)
1)
.921856
-.164159
-.112522
-.607131
-.171688
-.140218
-.161995
.100000
63967
41458
80878
03056
27675
91431
44008
00000
05455
60650
61094
26315
48834
54977
14657
00000
.500.LE.X.LE.
2.000
RATIONALAPPROXIMATION
IS R(6,6)
WITH PRECISION- 13.75 DIGITS
POO
POl
P02
P03
P04
P05
P06
QOO
Q01
Q02
Q03
Q04
Q05
Q06
-1)
0)
-1)
-2)
-3)
-4)
-6)
-4)
-1)
I)
n
2)
i)
i)
.190007
.145255
.812092
-.725683
.559368
-.321053
.996850
.383398
.950926
.175435
.827856
.117836
.609668
.100000
72856 75969
03101 02014
21046 70916
52768
64008
49168
83917
00706
03238
39262
65799
83566
97953
00000
34330
87053
10291
42953
66617
66574
79244
05180
14576
70546
00000
FIT TO E41(X)-EXP(-X)*P(X)/Q(X)
E*1(X)-EXP(-X)/X**5*P(Z)/Q(Z)
; Z-l/X
IN THE RANGE 2.000.LE.X.LE.10.000
IN THE RANGE 0.000.LE.Z.LE.
.100
RATIONALAPPROXIMATION
IS R(4,8)
WITH PRECISION- 13.93 DIGITS
RATIONALAPPROXIMATION
IS R(5,7)
WITH PRECISION- 14.84 DIGITS
POO
POl
P02
P03
P04
QOO
Q01
Q02
Q03
Q04
Q05
Q06
Q07
Q08
2)
2)
2)
0)
-6)
1)
3)
4)
4)
4)
4)
3)
2)
D
.353537
.293820
.145436
.999939
.520751
-.223458
.204802
.183199
.333915
.276686
.128585
.297624
.295401
.100000
04817
63659
89922
71607
56693
69506
30991
07073
80415
23525
73754
44755
74603
00000
63005
48671
68919
90340
51248
60041
30036
94430
91970
75097
40614
03964
13003
00000
POO
POl
P02
P03
P04
P05
QOO
Q01
Q02
Q03
Q04
Q05
Q06
Q07
-5)
-3)
-2)
-1)
-1)
-2)
-5)
-3)
-2)
-1)
0)
1)
1)
1)
.395133
.204661
.346003
.204274
.211759
-.228430
.395133
.263931
.672752
.828969
.515915
.155462
.206789
.100000
17174
41023
84617
58543
92902
28733
17174
38600
62012
69188
51447
70935
80159
00000
71982
99156
74784
62413
12236
96219
71976
20275
18175
14699
03679
57096
55259
00000
In all three ranges, the function E{(x) computed by rational (k, I) fits in single
precision was compared with the multiple-precision master routines for 20,000
pseudo-random values of x and 1/x. The largest fractional error noted was 1.1 x
io-13.
Finally, the functions E[n(x) were calculated for 1 <y < 7, 0 < n < 7 and
pseudo-random values of x and 1/x according to Eq. (2.22). Comparisons were
made with the same functions calculated via Eq. (2.3) using the single-precision
routine DCADRE [10]. No significant discrepancies were found at the relative error
level of 1 X 10 ~12, and it was observed that rational minimax methods require
-150 times less computation time than does DCADRE. For the case y = 0, a
pre-existant library function based on [9] was used, although better methods are
known [3]. Agreement with Gussman's tabulated values [11] was total.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
454
M. S. MILGRAM
Table 3.5
FIT TO E51(X)-P(X)/Q(X)+P15(LOG(X))
FIT TO E51(X)-EXP(-X)*P(X)/Q(X)
IN THE RANGE 0.000.LE.X.LE.
IN THE RANGE
.300
RATIONALAPPROXIMATIONIS R(3,2)
WITH PRECISION- 15.30 DIGITS
POO
POl
P02
P03
(
(
(
(
3)
4)
2)
0)
QOO (
Q01 (
Q02 (
4)
3)
1)
.838743
.142760
.880503
.556431
.148423
.100211
.100000
38519
40312
85483
96858
35068
06179
00000
74414
21125
46258
95740
49911
41411
00000
.300.LE.X.LE.
2.000
RATIONALAPPROXIMATION
IS R(5,10)
WITH PRECISION- 15.44 DIGITS
POO
POl
P02
P03
P04
P05
QOO
Q01
Q02
Q03
Q04
Q05
Q06
Q07
Q08
Q09
Q10
-3)
0)
1)
1)
0)
-3)
-4)
-3)
l)
2)
3)
3)
4)
3)
3)
2)
D
.500522
.179471
.222234
.379468
.991769
.144274
.322199
.967191
.114091
.364550
.320690
.977517
.126115
.736669
.201906
.245286
.100000
58419
65122
03877
68767
17992
11935
80730
28566
98728
65926
08398
80646
41159
39202
35278
16349
00000
15039
08398
87114
06738
22935
15928
27793
38765
01536
78242
60978
43698
04314
94918
47405
83681
00000
FIT TO E51(X)-EXP(-X)*P(X)/Q(X)
E51(X)-EXP(-X)/X**6*P(Z)/Q(Z)
; Z-l/X
IN THE RANGE 2.000.LE.X.LE.10.000
IN THE RANGE 0.000.LE.Z.LE.
.100
RATIONALAPPROXIMATION
IS R(4,9)
WITH PRECISION- 14.90 DIGITS
RATIONALAPPROXIMATION
IS R(5,7)
WITH PRECISION- 14.05 DIGITS
POO
POl
P02
P03
P04
QOO
Q01
Q02
Q03
Q04
Q05
Q06
Q07
Q08
Q09
2)
2)
2)
o)
-6)
0)
3)
4)
4)
4)
4)
4)
3)
2)
1)
.185231 13891 15313
.169071 89408 09323
.163822 45019 47477
.999973 45809 73645
.178967 24703 86040
-.332104 62172 40376
.127144 45160 39896
.223717 64184 09195
.632007 63231 39911
.803518 43255 59348
.632932 10036 44392
.259400 04237 29686
.479992 93424 76704
.373802 66825 57554
.100000 00000 00000
POO
POl
P02
P03
P04
P05
QOO
Q01
Q02
Q03
Q04
Q05
Q06
Q07
-6)
-4)
-2)
-2)
-3)
-4)
-6)
-4)
-2)
-1)
0)
1)
1)
1)
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
.854783
.624139
.138649
.948336
-.423100
-.370718
.854783
.803643
.279890
.462603
.380244
.147127
.229031
.100000
10464 47894
17717 77879
42989 29153
88932 79712
66195 40255
93388 82177
10464 47817
62915 37229
57603 97328
61191 05448
88250 95635
35379 71468
45317 90874
00000 00000
THE GENERALIZED INTEGRO-EXPONENTIAL FUNCTION
455
Table 3.6
FIT TO E61(X)-P(X)/Q(X)+P16(L0G(X))
FIT TO E61(X)-EXP(-X)*P(X)/Q(X)
IN THE RANGE 0.000.LE.X.LE.
IN THE RANGE
.100
RATIONALAPPROXIMATION
IS R(2,2)
WITH PRECISION- 15.16 DIGITS
POO
POl
P02
QOO
Q01
Q02
(
(
(
(
(
(
4)
5)
3)
5)
3)
1)
-.828583
.145297
.486899
.148343
.545404
.100000
83172
02796
73737
42996
94731
00000
51790
48708
68510
06228
07867
00000
.100.LE.X.LE.
2.000
RATIONALAPPROXIMATIONIS R(6,12)
WITH PRECISION- 14.72 DIGITS
POO
POl
P02
P03
P04
P05
P06
QOO
Q01
Q02
Q03
Q04
Q05
Q06
Q07
Q08
Q09
Q10
Qll
Q12
-3)
-1)
0)
1)
1)
0)
-3)
-7)
-2)
0)
1)
3)
3)
4)
4)
4)
4)
3)
2)
1)
.271478
.195102
.321565
.138382
.205248
.989908
.167302
.771598
.168613
.219806
.810987
.114098
.688229
.205074
.325821
.282942
.130505
.296709
.296971
.100000
83310 87788
35413 21396
23084 02674
85866
36427
76978
21625
59706
64843
16317
78546
17140
02344
91268
03449
94513
73635
78069
60052
00000
93074
90331
43590
00549
76959
91203
34924
02619
23387
19855
54280
06946
61645
81354
04683
15741
00000
FIT TO E61(X)-EXP(-X)*P(X)/Q(X)
E61(X)-EXP(-X)/X**7*P(Z)/Q(Z)
; Z-l/X
IN THE RANGE 2.000.LE.X.LE.10.000
IN THE RANGE 0.000.LE.Z.LE.
.100
RATIONALAPPROXIMATION
IS R(7,7)
WITH PRECISION- 14.29 DIGITS
RATIONALAPPROXIMATION
IS R(6,7)
WITH PRECISION- 14.98 DIGITS
POO
POl
P02
P03
P04
P05
P06
P07
QOO
QOl
Q02
Q03
Q04
Q05
Q06
Q07
D
-2)
-3)
-4)
-6)
-7)
-9)
■11)
0)
1)
3)
3)
3)
3)
2)
1)
.109840
.628986
.352279
.166654
.631863
.177898
.327229
.292513
.174130
.789054
.294284
.980479
.851988
.259397
.293770
.100000
75266
68217
00224
15521
11472
75053
37007
40388
46755
90506
05239
98971
71787
92046
63084
00000
90525
39343
82096
17304
37745
81043
14598
70733
89622
24093
39408
36415
21233
62330
51261
00000
POO
POl
P02
P03
P04
P05
P06
QOO
QOl
Q02
Q03
Q04
Q05
Q06
Q07
-6)
-5)
-3)
-2)
-3)
-3)
-4)
-6)
-4)
-3)
-2)
0)
0)
1)
1)
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
.117928
.887532
.210604
.154902
-.627848
.202298
-.409048
.117928
.121773
.487180
.965568
.100200
.527459
.125439
.100000
10446
43497
43465
34850
92801
30636
68342
90701
71136
37132
49091
53793
59744
68896
10446 90700
11274 90613
40530
34655
50733
47929
02841
00000
95863
76481
04699
77240
63380
00000
456
M. S. MILGRAM
Acknowledgments. The author is grateful to J. Blair for many helpful discussions,
to S. Jurgilas for aid with the REMES2 runs, and to R. Davis for demonstrating that
Eq. (2.23) can be evaluated in FORTRAN for arbitrary values of j. The referee
pointed out the existence of [26] from which the further reference to [11] was found.
Appendix A
A Short Table of Integrals
From the G-function representation (2.7) and integrals listed in [21], it is possible
to generate a table of very general integrals, which contain many useful results as
special cases, all of which are believed to be new. A short list follows.
J/.00
dtt^(at)Ek(ßt)
n
(A.l)
= fl--r-l/7>+2.*
P
+2
-y,l - v - y,...,l
^
Lr/+*+3,y+*+3l
n
0,p-
l,...,p-
- v - y;p,...,p
l;-v -y,...,-v
\
- y)'
Special case (j = -1)
f° fe-a,E^(ßt) dt
(A.la)
0_T_!V
~ß
r(l
+ y + /)/
1
\k+1l-a\'
g0 r(i + /) [7T7T7J
(t)
...
,
lol
lf|a|<l/?|-
Special case (k = y = 0, v = a = ß = 1, p = m; [20], [24])
/
Ex(t)El,(t)dt
Jo
(A.lb)
(-l)V/l
v¿ (-1)'
y~l> (m
l2,to
(A.2)
/
r^(/30^
•'o
/! ^2
= /3-1'-Ir(l
*'m-*'(f
+ log2 .
+ Y)(i, + Y)-
r ty-\t + ßy°EJ(zt)dt
(A.3)
1 - y;?,...,?
ÊULrJ+}-1
a - y,0,f - l,...,f
- 1;
Special case (y = 1, a = 1 + v,j = 0)
(A.3a)
/
(t + ß)-'-lEp(zt)dt
Jq
= ß->txp(ßz)Ev(ßz)/v.
Special case (a = 2, y = 2 — v,j = 0)
(A.3b)
(A.4)
fX tl-"(t
+ ß)'2Ev(zt)
dt = exp(/3z)z"1r(2
c4
a,ß
Y
wt \tsEf(zt)
z-S
T(y)
+4,
w T(a)T(ß)
J + Xj + 4
- v)E2_y(ßz),
dt
0;y - I,»' + 8,...,v + 8
8,v + 8 - l,...,v + 8 - l,a - l,ß - 1;
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
THE GENERALIZED INTEGRO-EXPONENTIAL FUNCTION
457
Special case (a = ß = 8 = 1, y = 2)
roc
(A.4a)
1
jf
log(l + wt)Ej(zt)dt
/
0;1 + v,...,l
0,0,v,...,v;
= -Gj:ij+3\[
+ v
Special case (ß = a - \, y = 2a, 8 = a - 1 - v)
i" [1 + y[ÍTw~t\l-2ata'l'"EÍ(zt)
dt
Jn
(A.4b)
_z^l-"(a-\)
31 I z_0;2a-l,a-l,...,a-l
a —1 — v, a —\, a —2,..., a —2;
Jj + 2,j + 3
WV7T
(A.5)
/
cos(at)Ej(ßt)dt
Jn
\;(v+l)/2,...,(v+l)/2
~
a
UJ+2-J+2\
2~'-X y I
ß ,to\'/2
(A.5a)
a2
1
+ 'i
i,("-i)A-.("-i)/2,
7 + 1/'
«
-) - ; - 1
2
(A.6a)
aE00
/32 /fol(l + ")/2 + //
if H<|j8|,
\/32
(A.6) r Sm(at)EJ(ßt)dt= ^lai^
'——GtfjJ
Jo
2
-a
0;(* + l)/2,...,(*'
.1Ê.
£ 0,(v~l)/2,...,(v-l)/2;
+ l)/2
\ a
' + 1/
„2-
if |a|<|/3|.
\ j8
f°t-(t-l)'-ß-lEJ(zt)dt
(A.7)
Ji
;s,...,s,a
= T(a-ß)Gf:lf+3[z 0,ß,s-
l,...,s-
1:
Special case (s = -/?, a = 1 — /, ß = I)
/•CO
/
t'-xEln(zt)
Ji
(A.7a)
=
T(n + 1)
,n+\
dt
1 ~j
k
j
E fr«.-£-+2-*-/(')+^+i_,(*)E ^,m(^7
: —0
m= 1
/
m -1
m= \
/= 0
- L
L (n - iy-miCmE{(z)
if /# n,/7 + 1,
/•OC
/
Ji
(A.7b)
t"'lE¿n(zt)dt
r(n + 1)
.« + 1
A;= 0 "•
m-1
For the special case / = n + 1, see Eq. (2.12).
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
458
M. S MILGRAM
Applied Mathematics Branch
Atomic Energy of Canada, Limited
Chalk River, Ontario. Canada KOJ 1J0
1. M. Abramowitz
& I. Stegun, Handbook of Mathematical Functions, Chapter 5, U. S. National
Bureau of Standards, Washington, DC, 1964.
2. I. K. Abu-Shumays,
Transcendental Functions Generalizing the Exponential Integrals, Northwestern
University (unpublished) report COO-2280-6,1973.
3. D. E. Amos, "Computation
of exponential integrals," A CM Trans. Math. Software, v. 6, 1980, pp.
365-377.
4. L. Berg, "On the estimation of the remainder term in the asymptotic expansion of the exponential
integral," Computing, v. 18, 1977, pp. 361-363.
5. B. S. Berger,
Tables of Zeros and Weights for Gauss-Laguerre Quadrature to 24S for n = 400, 500.
600, Dept. of Mechanical Engineering, Univ. of Maryland, College Park, MD. (Unpublished report.)
6. W. F. Breig & A. L. Crosbie. "Numerical computation of a generalized exponential integral
function," Math. Comp., v. 28, 1974,pp. 575-579.
7. R. P. Brent,
"A FORTRAN
multiple-precision
arithmetic package," ACM Trans. Math. Software,
v. 4, 1978,pp. 57-70; ibid. pp. 71-81.
8. S. Chandrasekhar,
Radiative Transfer, Dover, New York, 1960.
9. W. J. Cody & H. C. Thacher, Jr., "Rational Chebyshev approximations for the exponential integral
£,(.¥)." Math. Comp., v. 22, 1968, pp. 641-649.
10. DCADRE, IMSL Library, 6th Floor. GNB Bldg.,7500 Bellaire Blvd., Houston, TX.
11. E. A. Gussman, "Modification to the weighting function method for the calculation of Fraunhofer
lines in solar and stellar spectra," Z. Astrophys., v. 65, 1967, pp. 456-497.
12. H. C. Van de Hülst, "Scattering in a planetary atmosphere," Astrophys. J., v. 107, 1948, pp.
220-246.
13. H. C. Van de Hülst, Multiple Light Scattering, Vol. 1, Academic Press, New York, 1980.
14. D. R. Jeng.
E. J. Lee & K. J. de Witt,
"Exponential
integral kernels appearing in the radiative
heat flux," Indian J. Tech.,v. 13, 1975,pp. 72-75.
15. J. H. Johnson & J. M. Blair, REMES2: A FORTRAN Programme to Calculate Rational Minimax
Approximations to a Given Function, Atomic Energy of Canada Ltd., Report AECL-4210, 1973.
16. C. Kaplan,
On a Generalization of the Exponential Integral, Aerospace Research Lab. Report
ARL-69-0120, 1969; On Some Functions Related to the Exponential Integrals, Aerospace Research Lab.
Report ARL-70-0097, 1970; Asymptotic and Series Expansion of the Generalized Exponential Integrals, Air
Force Office of Scientific Research Interim Report AFOSR-TR-72-2147,1972.
17. J. LE Caine,
A Table of Integrals
Involving the functions
En(x),
National
Research
Council
of
Canada Report NRC-1553, 1945, Section 1.6.
18. Y. L. Luke,
The Special Functions and Their Approximations.
Academic Press, New York, 1969.
19. A. S. Meligy & E. M. El Gazzy, "On the function E„(z)," Proc. Cambridge Philos. Soc, v. 59,
1963, pp. 735-737.
20. M. S. Milgram,
"Approximate
solutions to the half-space integral transport equation near a plane
boundary," Canad. J. Phys., v. 58, 1980,pp. 1291-1310.
21. M. S. Milgram, "Some properties of the solution to the integral transport equation in semi-infinite
plane geometry," Atomkernenergie, v. 38, 1981, pp. 99-106.
22. M. S. Milgram, "Solution of the integral transport equation across a place boundary," Proc.
ANS/ENS
International
Topical Meeting on Advances in Mathematical
Methods for the Solution of Nuclear
Engineering Problems, Munich, FDR, (1981), pp. 207-217.
23. W. Neuhaus & S. Schottlander,
"The development of Aireys converging factors of the
exponential integral to a representation with remainder term," Computing, v. 15, 1975, pp. 41-52.
24. M. A. Sharaf, "On the A-transform of the exponential integrals," Astrophys. and Space Sei., v. 60,
1979, pp. 199-212.
25. R. R. Sharma & B. Zohuri,
"A general method for an accurate evaluation of exponential integrals
Ex(x), x > 0," J. Comput.Phys., v. 25, 1977,pp. 199-204.
26. A. Stankiewicz,
"The generalized integro-exponential functions," Acta Univ. Wratislav., No. 188,
1973, pp. 11-42.
27. I. A. Stegun
& R. Zucker,
"Automatic computing methods for special functions. Part II," J. Res.
Nat. Bur. Standards, v. 78B, 1974,pp. 199-218.
28. H. Strubbe, "Development of the SCHOONSCHIP program," Comput. Phys. Comm., v. 18, 1979,
pp. 1-5.
29. R. Terras,
" The determination of incomplete gamma functions through analytic integration," J.
Comput. Phys.. v. 31. 1979, pp. 146-151.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use