Further
Approximations
for Elliptic Integrals
By Yudell L. Luke*
Abstract. The present paper develops approximations for the three kinds of elliptic
integrals based on the Padé approximations for the square root. The work includes and
extends our previous work on the subject to provide efficient approximations over a larger
part of the complex k and <¡>planes.
1. Introduction. In previous work [1], [2], we gave some approximations for the
three kinds of complete and incomplete elliptic integrals which were derived by
using Padé approximations for the square root. The expansions are valid in a large
sector of the complex plane and are far more powerful than the analogous representations based on the binomial expansion for the square root. Nonetheless, the convergence properties of our previous developments weaken for k2 sin2 </>near unity.
In this paper, we develop new representations like those of [1], [2] to correct this
deficiency. The present study is suggested by analogous work for power series
developments of the first two kinds of elliptic integrals. In this connection, see the
recent study by Van de Vel [3] and the references quoted there. For convenience
and completeness, we also give the representations previously found. Our proofs are
very sketchy as the details are akin to those previously used.
2. Approximations for the Square Root and Other Data.
(1)
(1 - z)-1'2 = (2n + I)"1 ¿
(2)
6,(1 - z sin2 ft.)"1 + VnOz) ,
(1 - z)1'2 = 1 - 2z{2n + I)"1 ¿
sin'°",
77.-1 1—2
(3)
em = 1
if m = 0 ,
ím = 2
(4)
6m = m7r/(2ft + 1) .
+ WAz) ,
COS 6n
if m ^ 1 ,
Let
t _ 2-
(5)
2 ±2(1
-
z)112
where the sign is chosen so that \el\ lies outside the unit disc which is possible for all
z except z ^ 1. Then
4e-(2n+3/2)f
(6)
VAz) = —2
z1/2(l -
e-f)[l
+ e
-(2n+l)¡--|
'
Received July 30, 1969.
AMS Subject Classifications. Primary 33.19, 41.17.
Key Words and Phrases. Elliptic integrals, approximations
by rational functions.
* This research was supported by the U. S. Atomic Energy Commission under Contract
AT(11-1)1619.
191
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No.
192
YTJDELL L. LUKE
¿<2(l
(7)
Wniz) =
_ e-r)e-(2«+i/2)f
[1 _ e-C2»+l)f]
and for z fixed,
(8)
lim \Vn(z) and Wniz)] = 0 ,
2 ^ 1, |arg (1 - z)[ < v .
n—*oo
It is also convenient
to have representations
like (1) and (2) with z replaced by
-z. Thus
(9)
(1 + z)-1'2 = (2n + I)"1 ¿
6m(l + z sin2 6A~l + Rniz) ,
771=0
1 V-»
(10)
(1 + z)1'2 = 1 + 2z(2n + l)-1 ¿\
-2-
Sin
Ott.
i 1 + z cos 6m
i
C7 / \
+ Sniz)
For the error terms, let
1/2
, = 2 + z±2(l-rz)
(11)
2
where the sign is chosen so that \ev\ lies outside the unit disc, which is possible for all
2 except —1 2: z ^ — co. Then
(12)
Rniz) =
(13)
<S„(z)
Ae-(2n+3/2)7)
77
2,/2(l
+ e-')[l
ZW»(1 +
-
e-<2"+"'] '
e-,)e-(2n+l/2),
(2n+l)7,,
[1 + c-l«H-i»]
'
and for z fixed,
(14)
lim {Rniz) and Sniz)} = 0,
z j¿ -1,
|arg (1 + z)| < *-.
n—»00
Finally, we have need for some integrals:
(15)
Hid,a2)= f--f^l"J«l-o
2s
(16)
-1
H (6, a ) = c
arc tan (c tan 0) ,
|arg (1 — a )| < ir,
(17)
H ((9,a2) = (2g)-1 In
H(6,a)
(18)
a
c = (1 — a 24,1/2
)
|arg (1 ± ¿ctan0)|
1 4- grtan 0
1 — ¡7tan 0
< ir,
g = (a — 1 )
¿ = (— 1)
1/2
, g tan 0 real .
= tan0
if a = 1 ,
= ir/2e
if 0 = tt/2 and |arg (1 — a )\ < it ,
= 0
(19)
sin
if 0 = ir/2 and a2 > 1 .
/ -Sm22 " , da = a"2[ff (0, a2) - 0] .
•'o 1 — a sin a
In the above, the integrals
are to be interpreted
in the Cauchy sense when a2 sin2 0
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FURTHER
> 1. Indeed,
appropriate.
/oa\
(20)
ría
APPROXIMATIONS
throughout
2s
this paper,
f"
FOR ELLIPTIC
we interpret
cos ada
["
2s
,„4-i,
= (2c)
1 +
In-^—
c sin 0
1 — c sin 0
(21)
,
integrals
■2 .
((1
1 —
- a« )) sin a
c = (1 - a2)1'2,
| arg (1 — a2) | < ir ,
c sin 0+1
c sin 0 — 1|
(22)
J(6, a2) = (2c)_1 In
(23)
J(d, a2) = sin0
(24)
J(0, a2) = g~l arc tan (¡5sin 0) ,
3. Approximations
in this sense when
C°S ada
Ji6, O) = J -2
, 2 ■ 2 = /-2s
J o COS
cos a + a sin a
'i 1na
J(6,a)
193
INTEGRALS
csin0
j¿ 1
| arg (1 ± c sin 0) | < it.
c sin 0 real
if a2 = 1 .
g = (a2 — 1)1/2,
g sin 0 real.
for the Elliptic Integrals of the First and Third Kinds. Let
Fi4>, k, v) = /
(1 - v2 sin2 a)-1 il - A sin2 a)-1/2o
(25)
v sin <j>9e 1 ,
|arg (1 — fc )| < ir ,
|arg (1 — k sin2 <£)| < tt .
Using (1) and (15) — (18), we get
n+,k,,) -f£$
%Ál - 5 sin2öm)_I
2/c
(26)
Qni<t>,k,v)
=
Qn(<¡4,
fc,P) =
/
•A4sin 6mHj4>, k sin 0m)
(2ft + l)v*¿A
1 -
ik2/v2) sin2 0„
(1 — i-2 sin2 a)
^„(/c2
sin2 a)da
2c-(2"+1)r tan <b
.
:^
+ 0(ft-2)]
+ 0(e-4nr)
(1 - *<2sin2 <p)(4ft + 3) L~ ' 4ft 4- 3
Xi = 1
cosh f — 1
2iv sin 0 (cosh f — 1)
sinh f cos <j> (1 — v sin 0) sinh f
0 ^ an odd multiple of
(27)
0.(t'*'')-[
2tt sinh f
(4ft + 3) (cosh f
+ 0(e-ter)
" 1/2
_(2„+i)f
r
,
_ (2 - fc2)(cosh r + 1)
1
y2(coshr-l)
Ssinhr
" 2 + (i_,2)sinhf:
where ef is defined by (5) with z = k2 sin2 <j>.
Clearly for fixed </>,k and ?, under the conditions
given in (25),
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2 '
194
YUDELL L. LUKE
(28)
lim Q„(0, k,v) - 0.
In (21), the coefficient of Gi<t>,v2) can be replaced by (1 - k2/v2)~V2 - F„(fc2/V2)
provided |arg (1 — k2/v2)\ < x. This should be used only when the magnitude of
V„ik2/v2) does not exceed the magnitude of QA<i>,k, v), which happens approximately
when \es\ with z = k2 sin2 0 exceeds |ef| with z = k2/v2. In the case that 0, k, and v
are real, this happens approximately when 1 ;£ v2 sin2 0.
Some numerics illustrating these approximations and the striking realism of the
error formulations are given in my earlier works. It is clear that the accuracy
weakens for values of k2 sin2 <j>near unity. To correct this deficiency, we proceed
as follows. We can write
„,
.
s
/
b (0, k, V) - J
..„„s
secado;
y
2 • 2 w,
o (l — j7 sin
k'2 = 1 - A ,
. ,,2 ,
a)(l
+ k
tan
2~TÏ72 »
a)
|arg (1 + k'2 tan2 0)| < t .
Now use (9) and (20) - (24). Thus
Fi<b, k,v) =-l--r
¿-<Al-k'2sm2em)
(2ft +
1)(1
-
v2) 77-ti l -
2
V
(fc'2/(l
,2
-
Ji<t>,1 - «') Ta
(2ft + 1)(1 - y2) ' ST'
ä
„2)) sin2 em
'Si-
(fc'7(i
_ „»)) sin2
+ Sn{<t>,k,v),
tan
, ,
/" sec
sec aRnik
,k,v)
-- ali
S„(4>,
k,v) = J/-——■'o
°
11 — v sm
70Q4.
2e~(2n+1)vsin 0
&(*. *,-)-.
(31)
a)da
a
(1
I
p
"", ■ 2 "7" QsF1 - 4ft7^5
+ 0(n-2
— j/ sm 0) (4ft + 3) L
-f- «3
sin2 0 sinh g f 2(1 ~ y2)
"=1+cosh,-l
Ll-^sin2^1]-
il
+ 0(e-4""),
sinh 77
cosh
where e" is defined by (11) with 2 = fc'2tan2 <j>.Then under the same conditions as in
(29), with <b,k and v fixed,
(32)
lim Sni<t>,
k, v) = 0 .
71—»0O
The approximation (30) is convenient even when fc'2 tan2 0 > 1. Notice that if
0 -> tt/2 and fc -> 1 so that cos2 0 = fc'2, then fc'2tan2 0 -^ 1 and e"' -^3 -23/2
=0.172. The coefficient of ■/(</>,
1 — v2) in (30) can be replaced by
-,2[(1
-
,2)(fc2 -
V2)]-1'2 + V2(l -
V2A'Vnik'2/il
-
V2))
provided [arg (1 — fc'2/(l — v2))\ < -k. However, this should not be introduced
unless the magnitude of F„(fc'2/(1 — v2)) is approximately equal to or less than the
magnitude of Sni<l>,fc, v).
To get a useful approximation
when 0 is near 7r/2, we write
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FURTHER APPROXIMATIONS
(T/2)-«
F\^,k,Aj-F{4>,k,v)
= /o'
195
FOR ELLIPTIC INTEGRALS
Sec ada
a)(fc' + tan 2 a)\l/2
(1 - ¡/cos
tan 6) ' de
f* sec 0(1 + fc'21!
K ' i - v2+f- fc' tan
(33)
fc' tan i/- = cot 0 , y2sin 0 ^ 1 ,
[arg (1 + fc' tan ^)| < it .
Now use (10). We find that
Ki'/C) ") " F(*'fc'^ = (1 ~ v2)'1JV'(T~7))
(1
2s
X 1(34)
71
•
2(1 - v2)
'_) \p_sm
2ft +
1
2 n
0m_
777=11 — (1 — j,2) cos2 6mA
• 2 a
Bill
sin
0„
4r-4
+ 2ft + T¿íi
- (1
- v2) cos2 6„
(1 —
X /(-A, k'2 cos2 0m) + i¥n(0, fc, v) ,
Mn((f>,k,v) =
Mni<t>,k,v)
(35)
f* sec eSnjk'2 tan 6)d8
Jo
1 - v2 + fc'2tan20
2(cos 0/(1 + sin 0)) + (1 - sin 0)
(1 - fc sin 0r"(l
X
1+
4n + 1
<7= 2 sin 0—1
-/sin
0)(4ft+
+ 0(n2)
fc' sin 0
1 — fc sin 0
1)
/ COS0Y
U\l + sin 0/
'
2(1 — v ) sin 0
1 — v~sin2 0
In (34), the coefficient of J(ip, fc'2/(l — v2)) can be replaced by v(l — v2)*1—
(1 — v2)~lWn(l — v2) provided |arg v\ < ir/2. This replacement should not be used
unless approximately the magnitude of Wn(l — v2) is less than the magnitude of
Mn(4>, fc, v). If v and 0 are real, e_f = (1 — v)/(l + v) when 2 = 1 — v2. Thus, use
the alternative form provided
(1 - »0/(1 + v) t% (1 - sin 0)/(l + sin 0) ,
i.e., when sin 0 ^ v < 1. Under the same restrictions
as in (33), with 0, fc, and v
fixed,
(36)
lim Mni<t>,
fc,v) = 0 .
Our technique is applicable to derive an expansion relating K(k) = F{w/2, fc, 0)
and K(k'), where fc'2 = 1 — fc2,since it can be shown that
(37)
K(k) + w-*Qnk'2)Kik') = - -| /"* -&JË
7T •'O
sin2 0)(i0
(1 — fc' sin22axl/2
0)
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196
YUDELL L. LUKE
Thus
= 2(2« + I)"1 In 2 + 2 £ a„
K(k) + n'1 (hi k'2)K(k')
L
(38)
am = (1 -
An(k) = -—
(39)
+ -4„(fc),
In (1 + a,„)
774=1
A sin2 0m)1/2 ,
/
(lnsin20)r,i(fc'2sin20)d0,
ir •' u
^'^'VKf-ini-äMt^ii+oc.-.,]
+ 0
Hence with fcfixed, |arg fc| < 7r/2,
(40)
lim4„(fc) = 0.
71—700
4. Approximations for the Elliptic Integral of the Second Kind. Let
E(4>,
k)= f
(1 — fc sin a)
da
•7 n
(41)
|arg (1 — fc )| < ir,
|arg (1 — fc sin 0)| < n .
Here we use (2) and (15) — (19). Thus
n
E(4>, fc) = (2ft + 1)0 - 2(2n + I)"1 £ tan2 0^(0,
(42)
m=1
rn(0, fc) = /
Jo
T U IA
Wn(k2 sin2 a)da .
2e-(2"+1Ktan0(l-fc2sin20)
+ 0(e-ini)
-
-•
\
g.
. "1
,
2fc2sin2 0(cosh f + 1)
0"! =
fc2cos2 6m) + Tn(4>,k ),
,
.
sinh f
(cosh f - 1)
-Scos 0 sinh f
'
0 t^ an odd multiple of ~¡r
(43)
77f71" i\ _
f 2in;coshr - 1) T7'-cm-ik,.
H 2 ' V ~ ~L (4n+1) sinhfj
(J2
6
(1
(9fc2sin2 0 - 2) (cosh f - 1)
1
8 sinh f
2 '
k sin 0)
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FURTHER
APPROXIMATIONS
FOR ELLIPTIC
197
INTEGRALS
where ef is defined by (5) with z = fc2sin2 0. Under the same conditions as for (41)
with 0 and fcfixed
(44)
lim Tn(4>,fc) = 0.
7i—7(70
Numerical examples illustrating (42) and the remarkable
formulas for T„(0, fc) are given in [1], [2].
After the manner of getting (30), we find
,-,,,
,s
7, s .
, .
2fc'2
efficiency of the error
A sin2 0mJ(0, fc'2 cos2 8A)
i?(0, fc) = p„ik) sin 0 + r—-
2,-—2-
¿n -+■i m=i
1 — fc' cos 0m
+ Uni<t>,fc), |arg (1 + fc'2tan2 0)| < w ,
(45)
,,,
^k)
,
2fc'2 A
sin20m
= l-2n-+-l£il-
f*cos aS„ (fc'2tan2 a)da
?7„(0, fc) = /
Jo
rr /_. ;\
(46)
, ,
2ku
(l - fcVn+1
fc'2cos27„ = k + Y^u'
>
.
2e""<2n+I),(l — fc2SÍn20) SÍn0f
(4ft+ 1)-L1+iftTT
2 sinh 7?
cosh 1?+ 1
U=\T+k)
Ji_
_. _2s
. „7 _4n,x
+ 0(n )J + 0(e
}'
3 (cosh 77+ 1) cos 0
sinh 7?
'
where e" is defined by (11) with 2 = fc'2tan2 0. Thus for 0 and fc fixed and restricted
as in (45),
(47)
lim U«i4>,k)= 0.
n—700
In (45), we can take p„(fc) = fc provided that |2fcw/(l — w)| < |í/n(0, fc)|. In this
connection we notice that if fc and 0 are real, then (1 — fc)/(l + fc) < e-' when
fctan 0 > 1.
A straightforward
(48)
analysis shows that
E{k) - E(<t>,fc) = fc'2F(o;,fc,fc) ,
fc'tan w = cot 0 ,
and so we can apply the expansions previously developed as appropriate.
A relation analogous to (37) for the complete elliptic integral of the second
kind is
(49)
E(k) = ^E(k')
+ ^K(k')
- ^ {2 + In fc'2| f/2 ,.
-
J°
t'"!^
(l-fc'sin<0)
2
_ 2/V2 r/2sin20(lnsin20)4J0
T K
(1 -fc'2sin20)1/2
and from this one can derive the analogs of (38) — (40). The result would be rather
complicated and not as efficient as the known expansion for E(k) in powers of fc'2.
Rather than use the expansion which could be found from (49), a much more
efficient procedure would be to employ the Legendre relation,
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198
(50)
YUDELL L. LUKE
E(k)K(k') + Kik)Eik') - K(fc)K(fc') = x/2 ,
and the appropriate
expansions for K{k) and K{k').
Midwest Research Institute
Kansas City, Missouri 64110
1. Yudell
L. Luke,
"Approximations
for elliptic integrals," Math. Comp., v. 22, 1968, pp.
627-634. MR 37 #2412.
2. Yudell
L. Luke,
The Special Functions
and Their Approximations,
Vol. 2, Academic
Press, New York, 1969, pp. 269-273.
3. H. van de Vel, "On the series expansion method for computing
first and second kinds," Math Comp., v. 23, 1969, pp. 61-70.
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elliptic integrals
of the