Handbook of Elliptic Integrals for Engineers and Scientists
Engineers and physicists are more and more encountering integrations involving nonelementary integrals and higher transcendental functions. Such integrations frequently involve (not always in immediately re cognizable form) elliptic functions and ellipti
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Recurrence Formulas for the Integrals of the Twelve Jacobian Elliptic Functions. Integrals of odd powers of the twelve Jacobian functions, it is to be noted, are expressible solely in terms of Jacobian elliptic functions and more elementary functions, while the evaluation of integrals of even powers requires in addition the two functions E(u) and u.
A =f m
t m dt
V(1-t2)(1-k2t2)
=f sinmtp dtp
V1-k 2 sin 2cp
=fsnm u du
[t = sin q:> = sn u].
310.00
Ao = J du = u = F(q:>, k),
310.01
Al = J sn udu =-i- In (dnu-kcnu) = - ~ cosh-I [(dnu)/k'].
310.02
A2
=Jsn 2 udu = ~2- [u - E(u)] ,
310.03
A3
=Jsn3 udu =~ [kcnudnu + (1+ k 2 ) In (dnu - kcnu)].
[q:>=amu].
[E(u)
=
E(q:>, k)J.
2k
1 With all the following indefinite integrals, the constant of integration is to be understood. Moreover, for brevity we write E(u) for E(am u, k).
P. F. Byrd et al., Handbook of Elliptic Integrals for Engineers and Scientists © Springer-Verlag Berlin · Heidelberg 1971
Table of Integrals of Jacobian Elliptic Functions.
192
310.04
{A4
310.05 A
J sn 4udu=
=
_
2",+2-
3~4
[(2+k 2) u- 2(1+k2)E(u) + + k 2 sn u en udnuJ.
sn2m-lucnudnu+2m(1+k2)A2m+(1-2m)A2m_2 (2m+1)k2 •
_ sn·'" u cn u dn u + (2m + 1) (1 + k 2 ) A' m+1 - 2mA''''_1 310.06 A 2m+32(m+1)k2
B
m
-J
=
t"'dt -J drp V(t2 -1) (t 2 - k 2) sin"'cp V1- k 2 sin2rp
=Jns"'udu,
[t = 1/sinp = nsuJ.
311.00 Bo= Jdu=u=F(p,k),
Bl =Jnsudu=J~
311.01
snu
=
[p=amuJ. In [~_u_]. cnu
+ dnu
311.02 B2 = J ns 2 udu = u - E(u) - dnuesu,
311.03 Bs =Jns3 udu = ~ [(1 2
311.04 { 311.05 311.06
+ k 2 ) In (~_u_) cnu + dntt
[E(u)
=
E(p, k)J.
esu dnunsu].
B =Jns4udu =
B B
4
i
[(2
+k
_
u - 2(1
2m(1
2m+2 -
+k
2)
E(u) - dnuesu (ns2u
+ k2) B 2m + (1 -
+ 2 + 2k2)J.
2m) k 2 B 2m - 2 - en u dn u ns2m+lu 2m+ 1 •
(2m +1) (1+k 2) B 2m + 1 - 2m k 2 B 2m - 1- cn u dn u ns2m + 2 u 2(m 1)
+
2m+3 =
Cm -- - J
2)
tmdt - J eos"'cpdrp - Jen'" u V(1- t 2 ) (k'2-+ k 2 t2) V1- k2 sin2rp
d u,
[t = cos P = en u J.
312.00 Co = J du 312.01
=
u =F(p, k),
C1 = J enudu = [eos-1 (dnu)J/k = [sin-1 (ksnu)J/k.
[p=amuJ.
t 93
Integrals of the Twelve Jacobian Elliptic Functions.
2
312.02 C = J cn 2 udu =
;1 [E(u) -
[E(u) = E(rp, k)].
k'2 U ],
312.03 C3 = J cn3 udu = 2~8 [(2k2 - 1) sin-l (k sn u)
312.04
I
C4
+ ksnu dnu].
= J cn4 udu t
=
3/i'
[(2 - 3k2 ) k'2 U
+ 2(2k2 -1) E(u) + k2 snu cnu dnu].
312•05 C2m+2 = 2m{2/iI-t) CI ", +
(2m -t) /i'l C sm (2m
+ t) /i
-.+ snu dnucnl"'-lu
l
312.06 C2mH = (2m+t) (2/iI-t) Clm+1 + 2m/i'· Cam- 1 +snudnucnamu 2{m
D =j
'"
V(tl -
=j
tmdt
t){/i l + /i'S tl)
+ t) /il
dfll
V
cos m tp t - /ilsin·fII
=jnc"'udu
'
[t = 1/cosrp = ncu].
313.00 Do = J du = u = F(rp, k), 313.01 D1=
[rp
j ncudu= j -cnu- = -Ii' 1n [/i' sn cnu+ dn du
u
t
u ]
.
313.02 D 2 = Jnc 2 udu= /i~1 [k'2 u -E(u)+dnutnu], [E(u) = E(rp, k), rp
313.04
I
D4
= Jnc 4 udu =
3
/it"
[k'2(2k'2 - k 2) U
+ (2 -
4k2
= amu].
=
am u].
+ 2 (2k2 -1) E(u) +
+ k'2 nc2u ) tnudnu].
313.05
=
313.06 D
2m/iID. m _ 1 + (2m+ t) (t- 2/iI)D."'+1 +tnudnunc
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