
1544 SPECIAL FUNCTIONS AND THEIR PROPERTIES
30.13 Elliptic Integrals
30.13.1 Complete Ellipti c Integrals
◮ Definitions. Properties. Conversion formulas.
Complete elliptic integral of the first kind:
K(k) =
Z
π/2
0
dα
p
1 − k
2
sin
2
α
=
Z
1
0
dx
p
(1 − x
2
)(1 − k
2
x
2
)
.
Complete elliptic integral of the second kind:
E(k) =
Z
π/2
0
p
1 − k
2
sin
2
α dα =
Z
1
0
√
1 −k
2
x
2
√
1 −x
2
dx.
The argument k is called the elliptic modulus (k
2
< 1).
Notation:
k
′
=
p
1 − k
2
, K
′
(k) = K(k
′
), E
′
(k) = E(k
′
),
where k
′
is the complementary modulus.
Properties:
K(−k) = K(k), E(−k) = E(k);
K(k) = K
′
(k
′
), E(k) = E
′
(k
′
);
E(k)K
′
(k) + E
′
(k)K(k) − K(k)K
′
(k) =
π
2
.
Conversion formulas for complete elliptic integrals:
K
1 −k
′
1 +